I never give up.
For the past four seasons, I've started slow in at least one of my two longest-running leagues that I care about the most.
2005: 0-4, made it to 8-5, lost championship game (basically because Seattle wanted Alexander to set TD record)
2006: 1-3, made it to 8-5, lost first playoff game
2007: 1-3, made it to 9-4, won championship game
2008: 1-5, made it to 7-6, lost championship game
Last year was the most fortunate, as I had the DeAngelo Williams explosion carry a mediocre team to within a few points of a championship. In most of the other years, I just kept plugging away.
This year, I'm 0-4 with a team that has fought injuries to be in the middle of the pack in points, but I've played teams that have finished 2nd, 1st, 2nd, and 2nd in points. The rest of the league is "allowing" 61-77 points per week, and then there's my team allowing 93 points per week. There's not much you can do, but clearly I can't give up on this team. At some point, the scheduling breaks should even out, and if my team gets healthy, I think I still have a chance to go 7-6. By the way, in the other big league for me, I'm 3-1 with many of the same players (I don't usually have such similarities in rosters, it just worked out that way), with one of the differences being that my team is 7th out of 12 in points allowed.
Don't go down without a fight, but realistically assess your playoff odds as far as how long and how much to spend. If a 12 team league has four teams make the playoffs, that's one thing. But if it's a 10 team league where six teams make the playoffs, 0-4 can be overcome. Remember, unlike a real NFL team, your fantasy team doesn't have to worry about dissension in the locker room. All I know is if I give it my best shot and fall short, I can accept it whatever the reason. If I give up, and then see a player I didn't pick up explode down the stretch, then I'll always wonder what might have been had I not stopped trying.
By the way, every league should have at least some weekly top score prizes to keep teams who start slow with incentive and something to root for. A consolation playoffs is OK, but I'd rather have the weekly prizes.