TL;DR version: The players, DM, and system all contribute to 15MD individually, but usually their interaction is more important. None of these is a silver bullet, but in general robust systems should depend less on who is playing them, and should encourage the play you want to achieve. When the group is a known quantity, however, houserules and other table agreements do the trick. Finally, I discuss a momentum system I use, and ponder a "desperation" mechanic that would give incentive for characters to carry on while still giving them the feeling of lack of resources.
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I think the 15MD can be attributed to the interplay between players, DM, and system. It is possible to avoid it by changing any of these three, or any of the 3 interactions, but for quite different reasons.
Clearly D&D permits the 15MD at the system level. Without considering the players or DMs, one way to get rid of it is to simply not make it possible. Doing so could muck with the feeling or functionality of the game enough that some might not want to, but that is getting into interactions.
Next, there are the players. If they don't like, or simply never consider, the 15MD they can simply press on when they could rest. For most groups, however, the decision of whether to rest "early" is at least influenced by the system and the narrative. And at some level it has to be, because otherwise the PCs only stop when they're dead or incapacitated. (Some groups find might the "30MD" unpalatable, for instance, but eventually every group will reach the point where a rest seems necessary).
Finally, there is the DM. Regardless of player or system, if the DM wants to make resting impossible that is usually in his or her power. More likely, however, is that the DM wants to make it anywhere from inconvenient to suicidal, which will influence the players.
So, what are the basic interactions between these elements?
DM and Player: IMO, the fundamental interaction in a non-dysfunctional game is agreement that everyone is here for the fun. Besides general human respect, this usually requires some degree of player buy-in to the campaign, and (sometimes tacit) boundaries on what the DM will do. If either the DM or the players dislike the 15MD, that can be prevented here by outright agreement. The DM can also use more indirect means, so that resting is, as mentioned earlier, inconvenient or suicidal. If the narrative has player buy-in, they will respect this when it occurs. In short, for some groups, the system can totally allow the 15MD, yet it will never or rarely show up because of either the DM, the players, or (more likely) their interaction.
Why isn't this the silver bullet? Primarily, because all games are at least a little dysfunctional. I take this as axiomatic, like all people being at least a little dysfunctional. If there is conflict on this point (e.g. a player whose verisimilitude is ruined because he feels he's gimping himself when the rules would let him be powerful) then a compromise works, but might be unsatisfying. Or, sometimes, the compromise doesn't work and games implode. A secondary reason is that a DM might find narrative excuses to avoid the 15MD wearying or simply not right for the setting/story. Note that both examples above can come from the interaction of player/system and DM/system. In any case, a good game needs to cope with a little dysfunction, and sometimes the weight of that dysfunction can't be help supported solely by the DM/player relationship.
DM and system: I think this is usually the least problematic interaction, or so problematic the game doesn't exist. If there is a serious problem here, a different system is probably the best idea. Every system has some level of meta-narrative baked into it, and part of this is how creatures use their resources, different magic systems, how powerful creatures feel, etc. So, choose the right system and the 15MD is not a problem. And if the system is a little wrong, a simple houserule might fix it right up!
Why isn't this the silver bullet? I've yet to meet a DM that believed they had found the perfect system. The best one, perhaps, but not perfect. If everything else about some system is wonderful, but it allows the 15MD, then there will still be conflict here. Furthermore, players have system opinions as well (player/system interaction) and people want to play systems they like (negotiated in a player/DM interaction). Finally, even when houserules are a good option, they are their own point of contention in almost any game. Even when everyone agrees a houserule is necessary, the specific form it takes could be contentious. Finally, even though rule 0 exists, using it sometimes indicates there is a problem (or mismatch) with a system. Wise DMs use it judiciously, in my opinion, and player trust can be a factor.
Player and system: The players act within the rules (usually

), and as a good first approximation most attempt some degree of optimization within that system. And the system better be able to handle that. If a system allows the 15MD, many players sure as heck will want to use it when possible. (DannyAlcatraz wrote earlier that saving resources in combat is tactical, and smart play. Well, so is going nova if there is no penalty. The player/PC distinction might influence which one is favored, but even most characters would probably prefer the latter when possible. This is usually how the real world works: If I can have two tests in a day, or schedule them a week apart, most people will prefer to "nova both tests" if there are not other considerations.) In short, the system is a major influence on character behavior, and the primary way of mediating player impact on the world, at least in combat. As with DM/system interaction, then, we should find a system that encourages the player behavior we want to see.
Why this isn't a silver bullet: For much the same reasons it wasn't for the DM. Systems aren't perfect, there are conflicts with the other people in the group, and house rules can become their own problem.
What is my overall take? In general, I feel the mechanics of the game should support the narrative of the game, and vice versa. Thus, I think a strong game will encourage the players and DM to act in ways consistent with this narrative regardless of who the players and DM actually are. (If this isn't the game they want to play, that is totally acceptable.) In game design, I can only change the system, I can't choose the players or DM. Thus, the strongest option is robust rules that minimize reliance on quality player or DM intervention to work as intended. (I consider this akin to good industrial tolerances in manufacturing, or strong institutions in nations. Yes, anyone can want to drive safely, and some will even be able to achieve it, but if we have strong cultural and legal institutions with regard to driving then almost everyone will.)
At an individual game table, the answer might be different since the group isn't an unknown quantity. Truly robust rules that work for a wide swath of people are much more difficult than small houserules and "gentlemen's agreements", so when the latter work there is no reason to despise them. They also aren't a good replacement for better systems.
I play in a game right now that uses a momentum system, such that a large portion of a character's power comes as they succeed in combat. For example, a character's reserve of spell points is smaller than the amount of momentum (which can be spent as spell points) they will gain over a typical combat for quite a while. In this system, going nova depletes reserves, but it does diminish the 15MD.
However, the 15MD is still superior in this system when you can get it, since more spell points in reserve would always be preferable. I've been thinking about ways to add a new tension to the game. I think the trick is to allow characters to trade off the risk of being low on resources for advances or abilities that otherwise aren't available. Maybe a "desperation" mechanic. Such things crop up in a lot of games, but are usually not very good. For example, in 4e there are various abilities that grant a bonus when bloodied, but I think most are underwhelming unless you're a monster. What if one's ability to use powerful abilities actually increased as the player runs low on resources? It's less safe, but the stakes are raised and some achievements might only be possible when things get desperate. So, characters that use the 15MD can still succeed, but they don't uncover the most powerful reserves of power until they're against the ropes. The risk should be enough that, on balance, one wouldn't recklessly burn resources to get to that point. In short, you're not strictly better, but you are different.
In a 4e setting, I can see something like revamped milestones and accounting of healing surges (rather than hit points) as a measure for determining when these kick in. A player without any healing surges left might be an absolute terror, but they would be taking a pretty big risk. For some parties the 15MD would still be attractive solely for personality reasons, but if they feel a push to go onward because they are better by some metric they value (say more frequent critical hits and the ability to "overdrive" spells) then it would encourage taking a few more risks. In some sense, it is "momentum" at the encounter level, rather than the round level.
Any ideas along these lines, or other comments?