There are exactly two situations when you should fudge:
- When it makes the game more fun
- When your game is badly broken and there is no other good way to fix it.
Obviously the second guideline ties into the first, which should be the final arbiter for all of your fudging decisions, not just for dice rolls.
The only time you shouldn't (potentially) fudge is when you are demonstrating how your game world operates. If you want your players to know exactly what will happen if they don't make their rolls, you shouldn't fudge.
Remember, though, while you shouldn't hesitate to fudge as necessary, you should avoid fudging whenever possible. It is a means to an end, not a regular component of game play.
Two quick examples based on your original question:
A goblin archer crits a PC, causing the PC to die -- Wouldn't it be more enjoyable for everyone (player and DM) if the character didn't die at the hands of a low-level, non-boss cannon fodder monster? Absolutely, and letting them not die was a good decision so long as your players aren't getting the idea they are completely invincible (you are demonstrating death is something to be feared).
The entire game world will implode if the characters fail a roll -- The scenario you outline for the failed roll sounds amazing. As a DM, I would be tempted to play this out just for the sheer twists and turns the plot could provide. Unfortunately, as much as I believe games should have realistic outcomes even with fantasy scenarios, this kind of inflexibility (will implode vs. may implode barring PC intervention) and attaching of dire outcomes to random events (a single roll of the die) is bad design. The right time to fudge here would be if you are in the midst of play and you suddenly become aware of how much fun your players won't have dealing with their entire world being stomped on.
On the other hand, if you want the players to know their place in the world (demonstrating they make a huge impact on world events by simply speaking to someone), you may wish not to fudge the example diplomacy role. (It still is bad design, however, to do this type of thing on a regular basis - you have given up control of your game, and therefore control of the fun, to random luck.)
#Caveats
Fudging is an extremely useful tool any DM should keep in their toolbox. That said, there are a couple drawbacks to fudging which are well worth thinking about.
Frequent fudging is usually a sign of poor game construction -- While "bad" game construction is nothing to be looked down upon necessarily (we are all constantly learning), there are, objectively speaking, some design choices that lower fun more than others. If you are constantly correcting your game on the fly to make it fun, it would probably behoove you to re-examine how you are building things in the first place.
Fudging rolls can devalue character abilities in a player's mind or in real game terms -- Simply put, if your players believe their rolls are worthless (i.e. they have no control over their characters/stats or the game environment), they will have less fun. Likewise, if you are setting difficulty levels to a point where player stats and abilities barely make an impact, why are the players playing at all?
The good news is that there are simple fixes for both of these.
If you see a problem coming, fix it. The fact that you needed to ask in regards to whether or not you should fudge the example diplomacy roll means you have already identified an issue that warrants reviewing the scenario indicated and making some changes so that no fudging should be necessary.
If you do fudge, use a Magician's Force style vs. railroading (i.e. overtly saying "No, this is what happens..."). While fudging (or suspicion of fudging) may lessen the degree to which players trust your die rolls and their stats, if they can be led to believe that you are in fact not fudging (or are vaguely uncertain if you are), they will be much happier in the long run.
And just an FYI, your players will likely believe you are fudging, regardless of whether you do or not. I have had brand-new players suspect me of fudging simply because I didn't slaughter them at first level -- with no fudging on my part at all.
- Seek alternatives to roll fudging. Thinking about alternate scenarios (e.g. how can the players stop a global catastrophe if they do in fact lose this diplomacy roll) is the best method, but simply lowering the difficulty of a target number can go a long way. This allows the players to roll and utilize bonuses while vaguely setting things up in the players' favor.
Finally, here is a link to another answer I wrote regarding the issue of hidden stats for monsters and non-dice roll fudging. While it's somewhat indirect to the original question, it may clarify some questions regarding why adjusting (hidden) target difficulties is good practice in my opinion.
