Pendragon comes to mind as a great example of a game with plot as a premise. Beyond the many published scenarios (which I find mostly pretty spiffy), there is the grand campaign structure.
It has the advantage of drawing upon an incredibly rich myth cycle laden with multiple variations and interpretations. There is a similar resource in classical mythology, as suggested for instance in the work of Robert Graves. (ICE's Mythic Greece supplement for Rolemaster and Hero System is a nifty RPG resource.) Starting with the same familiar framework, two campaigns can produce very memorably different epic sagas.
That Providence or the Fates so often foretell tragedy in those milieus may be significant. With the prominent examples of doomed heroes, it may be easier to accept failure as a non-deterministic possibility. Gravitas and game may be in greater harmony than game and "happily ever after" story-telling expectations.
The thing I find problematic about Pendragon, and it applies to other campaigns as well, is that the PCs really don't make much difference in the end to how the campaign ends. They are peripheral to the story. The Fates have a very specific role of making sure things turn out the way things are supposed to. If you don't follow through with the story because of PC actions, it seems to rather miss the point of the story-line in not finishing it.
Not that alternatives are necessarily better. A purely "sandbox" campaign without an overarching plot often degenerates into players sitting around doing nothing until the GM puts a plot hook in front of them out of sheer frustration, or players who simply take the first plot hook every time without ever considering alternatives. Again in the end the world hardly notices the PCs existence, although in this case it's because they never do anything of real significance without the GMs intervention.
A third option is the campaign with a plot, where the GM has already assumed the players won't be peripheral to the main story, and has assigned them specific roles. This runs into problems as soon as the players start to deviate from the role the GM has assigneed them. You can either shift them back into the storyline, or ignore it, or let it go on without them interacting with it.
So that's three ways things can and do go wrong. A plot which the PCs are peripheral to is one they might resent if they start to feel they can't afffect the world. A plot which the PCs are an integral part of starts to go wrong when they decide to do something else. A lack of plot can lead to PCs who don't do anything until the GM tells them what to do. All these campaign styles can go wrong in other ways, but these are the ones that I've seen the most.
As for solutions, ultimately I think it's down to player expectations of what the campaign is about. Any of the three ways I've described can lead to awesome campaigns if that's what the players are looking for and if they and the GM deliver on those expectations. If the GM has made it plain what sort of campaign they''re running and the players go along with it, that should keep things going well.