There's a CRPG I've been meaning to play where this is basically the plot; there was a generation ship, it was heading towards some planet or another, but the social and political structure on the ship broke down at some point and now there's no one actually in charge, the ship is getting run down, and they probably blew past their destination a hundred years ago if they were even still on course at all.

I remember someone pointing out that a generation ship could be problematic because you have one generation who decides to launch this expedition but will never see the end, multiple generations who didn't choose this life and won't get to see the benefits, and then one generation who actually gets to the planet but might not even want to be there. Without some kind of cryogenic sleep or relativistic speeds the whole thing might fall apart just because most of the people involved "didn't sign up for this" but they have to toil away anyway for someone else to benefit from it.