In what context?
If you mean multiplayer, then I tend to stick through until the end, but I've been known to ragequit on occasion.
If you mean not finishing the campaign, then that happens when the game is terrible (which has only happened once or twice) or if I simply don't have enough time to play it, forget about it/lose interest over time and go play something else. Speaking of which, I still need to go back and finish Fallout: New Vegas...
If you mean something smaller (i.e. when replaying a level of a game), I'll quit once I've finished the good bits. For instance, there are certain segments of games which I find to be amazingly fun followed by decidedly mediocre segments (or segments of exposition which I don't feel like sitting through for the 10th unskippable time). Alternately, if I mess up badly enough, I'll just quit and possibly start over. This usually happens whenever I play Civ 5 and accidentally set it up wrong so that I can't win the way I was trying to--I spent HOURS trying to get a cultural victory only to find that I'd accidentally unchecked the box and couldn't win that way. That was frustrating.
I hope that this answers your question.