IntroductionI've noticed that a lot of people take two urns with them when fishing/mining/cooking whether they'll finish off the first urn or not. However, this isn't quite ideal, since you could have cooked an extra fish, mined an extra ore and fished an extra fish. I'd like to supply some details as to when to take one (partially full) urn rather than two.
The model to keep in mind is this table:
 Take one urn  Take two urns 
Fill no urns  Good!  Bad! 
Fill one urn  Bad(ish)!  Good! 
For the sake of concreteness, let's consider Mining and banking iron ore (say at Rimmington mine, so that you actually go to a bank every trip). Lets count how many ores we mine in each situation.
 Take no urns  Take one urn  Take two urns 
Fill no urns  28 ores, xp wasted  27 ores  26 ores 
Fill one urn  Impossible  28 ores, xp wasted  27 ores 
Since a round trip (from the mine to the bank and back to the mine) at Rimmington Mine takes about 27 seconds (using the Clan Vexillium), only getting 26 ores in a trip wastes some time relative to getting 27 ores: roughly a second.
Conversely, getting 28 ores gives you a second, but wastes potential experience from the urns.
In order to decide when to take one urn or two urns (or no urns at all!), we have to do some analysis.
MiningWe'll start with Mining. As noted above, our outcome table is
 Take no urns  Take one urn  Take two urns 
Fill no urns  28 ores, xp wasted  27 ores  26 ores 
Fill one urn  Impossible  28 ores, xp wasted  27 ores 
The only randomness in mining comes from the possibility of getting gems. Assuming you want to keep these gems the gem takes up an inventory space and hence affects the probability of filling an urn.
In my quest for 99 Mining, I mined (about) 299325 iron ores and got 536 gems (not counting those obtained by mining clay). This gives a probability of about 0.001791 (~1/558) for getting a gem for a given iron rock. Thus the probability of getting a gem in 26 iron rocks is about 0.04553 (~1/22)
The other numbers we need are simple. It takes 1625 experience to fill a Mining urn, or 46.43 iron mined, since iron gives 25 experience. Thus, 27 iron fills 945/1625 or 58.15% of the urn.
We can summarize this like so: if you get your partially full urn before the trip and the urn is
40% or less full  the urn will not be filled 
41% full  11/13 of the time this means the urn will not be filled (because of rounding). The other 2/13 of the time the urn will be filled after 27 iron are mined 
4243% full  The urn will be filled after 27 iron are mined 
44% or more full  The urn will be filled after 26 iron are mined 
We can use this to fill in the probabilities for the above outcomes. Note that this includes the possibility of getting a gem.
Given the choice of number of urns taken, either no urn or one urn will be filled. Thus each column should add to 1.
40% or less full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  1  1 
Fill one urn  0  0  0 
41% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  0.6824  1 
Fill one urn  0  0.3176  0 
4243% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  0.04724  1 
Fill one urn  0  0.9528  0 
4445% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  0.001092  0.04553 
Fill one urn  0  0.9990  0.9545 
4647% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  1.626E5  0.001013 
Fill one urn  0  1.000  0.9990 
48%:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  4.197E6  2.640E4 
Fill one urn  0  1.000  0.9997 
>48%:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  1.746E7  1.448E5 
Fill one urn  0  1.000  1.000 
There's always the possibility that you get a ton of gems and fill no urns, but this is incredibly unlikely.
We can now fill in the expected gain or loss from each choice. Recall that we're treating 28 ores/inv as 1 second gain and 26 ore/s inv as one second loss. Additionally, I'll treat not filling up an urn as a loss of 1 second per 14.7 experience wasted. (This assumes a rate of 53k xp/hour. Higher rates are possible with tick manipulation and mining at the wilderness location, but it doesn't really matter too much here.)
40% or less full:
 Take no urns  Take one urn  Take two urns 
Expected time wasted  12.3  0  1 
41% full:
 Take no urns  Take one urn  Take two urns 
Expected time wasted  12.3  0.1666  1 
(It's a negative expected value since you have a good chance of getting 28 ores, but you fill the urn at the last moment, so not much xp is wasted)
42% full:
 Take no urns  Take one urn  Take two urns 
Expected time wasted  12.3  0.3703  1 
43% full:
 Take no urns  Take one urn  Take two urns 
Expected time wasted  12.3  0.1762  1 
44% full:
 Take no urns  Take one urn  Take two urns 
Expected time wasted  12.3  0.03006  0.04553 
45% full:
 Take no urns  Take one urn  Take two urns 
Expected time wasted  12.3  0.2675  0.04553 
46% full:
 Take no urns  Take one urn  Take two urns 
Expected time wasted  12.3  0.4705  0.001013 
47% full:
 Take no urns  Take one urn  Take two urns 
Expected time wasted  12.3  0.6743  0.001013 
48% full:
 Take no urns  Take one urn  Take two urns 
Expected time wasted  12.3  0.9120  2.640E4 
TL;DR: Check how full your urn is before banking. If it's 44% full or less, just take the partially filled urn. If it's 45% full or more, take the partially full urn and a new urn. This can be accomplished by having a preset for each.
CookingHere our outcome table is
 Take one urn  Take two urns 
Fill no urns  27 food cooked  26 food cooked, space wasted 
Fill no urns  27 food cooked, xp wasted  26 food cooked

With cooking, urn management doesn't have as much effect since banking is so fast. In typical cases, banking takes 2 ticks (1.2 seconds) with presets. If you're lagging, this'll be longer. It may be shorter in NXT (never tried it). Under this assumption, taking only one urn is rarely advantageous. The loss due to only cooking 26 food is less than a twentieth of a second per inventory.
If you're lighting your own fires, it's much more worth it to have your two presets both with two urns, but one with a log in it as well. Manually taking a log out of the bank wastes much more time no matter how you do it.
1) Deposit the cooked food you have and pull out both the log and raw food.
2) Pull out the preset, then open the bank again and replace a raw food with a log.
3) Have a third preset with the log (costs 533 Runecoins or 3 Bonds) and open the preset menu to pull out the third preset.
All three of these take enough time that the time saved from urn management is not worth it.
If you aren't lighting your own fires (BTW, is it supposed to be possible for IM to cook on other people's fires?) then you can do urn management. I don't suggest this though, since either the fires aren't lit very fast, or there's a crowd around it, which makes cooking difficult.
If you do decide to do urn management, you should know that 26 trout fill up 38.3% of the urn (so you should pull out 2 urns if the urn is 61% full or higher). 26 salmon fill up 49.3% of an urn so you should pull out 2 urns if the urn is 50% full or more.
Urn management was much more useful for the Fourth of July Event, since banking actually took a bit of time and there were no fires to worry about.
TL;DR: don't bother with urn management when cooking. Instead, use one of your two presets to make taking a log out much faster.
FishingAssumptions for this section:
1. You're fishing at the Combat Academy (parts of this analysis also apply to the Gunnarsgrunn location during the Fourth of July Event)
2. The ratio of trout to salmon is 3:2 on average.
3. Fishing experience is valued at 60k/hour. This number is lower at lower levels, but it doesn't matter very much for these calculations.
4. Your fishing urns are always filled by multiples of 10 experience.
5. It takes 2 ticks to bank with presets.
6. There are 8 possible locations for the fish to appear. Does anyone know if the fish can appear in the middle square of the north locations? Right now all I've seen are these locations: XXOXX.
7. You always fish from the northernmost location available (obviously you don't want to move if you don't need to; Make your decision when you come back from the bank)
8. Jagex hasn't fixed the bug where you can't fish if you have 1 feather and a full inventory.
Under these assumptions, the average number of ticks to get to the bank is (7*20+6*21+5*22+4*22+3*24+2*25+1*25+0*26)/(7+6+5+4+3+2+1+0) = 21.821 ticks. (Note that under our assumptions, you'll never use the southernmost fishing location, since the other location will be farther north). Hence, a round trip takes on average 2*21.821+2=45.643 ticks=27.386 seconds.
If we take 26 fish as the default, then being able to fish an extra saves you 1/26 of a trip, or 1.053 seconds. Fishing fewer fish than 26 has the opposite effect.
We can also give value to the experience wasted by not using an urn. Fishing is roughly 60k xp/hour, or 16.667 xp/second, so for every 83.333 experience NOT put into an urn, you waste one second worth of fishing training.
Our outcome table is thus
 Take no urns  Take one urn  Take two urns 
Fill no urns  +1 fish, urn xp  +0 fish, normal urn xp  1 fish, normal urn xp 
Fill one urn  Can't happen  +1 fish, some wasted xp  +0 fish, normal urn xp 
In terms of time wasted (or gained),
 Take no urns  Take one urn  Take two urns 
Fill no urns  17.307  0  1.053 
Fill one urn  0  Some Amount1.053  0 
"Some Amount" depends on how much xp is gained after the urn is teleported.
Given how full the urn is at the start, we can calculate the probability that the urn is filled during that trip.
<27% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  1  1 
Fill one urn  0  0  0 
27% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  1.000  1 
Fill one urn  0  3.934E13  0 
28% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  1.000  1 
Fill one urn  0  1.1913E10  0 
29% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  1.000  1 
Fill one urn  0  4.624E9  0 
30% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  1.000  1.000 
Fill one urn  0  7.145E8  2.085E11 
31% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  1.000  1.000 
Fill one urn  0  9.309E7  7.683E10 
32% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  1.000  1.000 
Fill one urn  0  1.777E6  1.595E8 
33% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  0.9999  1.000 
Fill one urn  0  1.185E4  2.621E7 
34% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  0.9994  1.000 
Fill one urn  0  6.168E4  6.728E6 
35% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  0.9970  0.9999 
Fill one urn  0  0.002966  5.196E5 
36% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  0.9856  0.9997 
Fill one urn  0  0.01438  3.061E4 
37% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  0.9585  0.9984 
Fill one urn  0  0.04151  0.001650 
38% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  0.9002  0.9908 
Fill one urn  0  0.09977  0.009191 
39% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  0.7799  0.9711 
Fill one urn  0  0.2201  0.02894 
40% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  0.5892  0.9250 
Fill one urn  0  0.4107  0.07503 
41% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  0.4012  0.8220 
Fill one urn  0  0.5988  0.1780 
42% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  0.2302  0.6429 
Fill one urn  0  0.7698  0.3571 
43% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  0.09004  0.4539 
Fill one urn  0  0.9100  0.5461 
44% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  0.02754  0.2711 
Fill one urn  0  0.9725  0.7289 
45% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  0.006805  0.1120 
Fill one urn  0  0.9932  0.8880 
46% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  0.001085  0.03576 
Fill one urn  0  0.9989  0.9642 
47% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  2.640E5  0.009198 
Fill one urn  0  1.000  0.9908 
48% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  0  0.001525 
Fill one urn  0  1  0.9985 
49% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  0  3.096E5 
Fill one urn  0  1  1.000 
>49% full:
 Take no urns  Take one urn  Take two urns 
Fill no urns  1  0  0 
Fill one urn  0  1  1 
I'm going to skip showing the expected time wasted for each of these, since that'll take a lot of space. Instead I'll skip right to the TL;DR
TL;DR:Check your urn before banking. If it's 43% full or more, take out 2 urns. Otherwise take out 1 urn. Your preset with 1 urn should have at least 27 feathers because there's a small but significant probability that you'll teleport that urn and be left with 27 spaces (plus feathers). The other preset should have at least 26 feathers.
(btw, sorry for those of you who saw my earlier recommendation. I severely overestimated the cost of teleporting an urn and finishing the inventory with no urn. It turns out that the low probability of this happening combined with the small amount of xp wasted makes it worth it to only take one urn up to much higher percentages.)
Urns and deposit boxes/StilesIt's generally worth it to take urns with you when banking iron at the Dwarven Mine Resource Dungeon and when fishing on Karamja. However, if you take too many urns, you'll end up making many trips to the deposit box/Stiles with little benefit. However, if you take too few you'll end up having to bank to get more urns more often. I've set up a model for this, and for plausible numbers, it's most beneficial to take 812 urns in both situations.
If anyone would be kind enough to provide more precision, I'd be happy to calculate the ideal number of urns for each. Hecate claims that taking 10 urns is the best for Dwarven Mine banking, but I don't know his methods.
Numbers I'd need:
Average number of ores/fish per hour NOT counting banking/noting time. (=Number_per_hour/(1trip_time*trips_per_hour)
Average time to make a trip to the deposit box/Stiles and back.
Average amount of time it takes to get urns and come back.
ConclusionI know my methods are fairly crude, but I hope you're convinced that you can save yourself time by paying attention to how full your urns are. Feedback is welcome and suggestions for improving the formatting of the tables is very welcome.