It's very unlikely.. We're 10 points behind 4th currently, however we do have a game in hand, but we can't be sure we'll win that, with the way we've been playing this season.
We'd also need Newcastle, Chelsea & Arsenal or Spurs to all mess up, and even if that did happen, we still have to string together a set of wins, which I can't see happening.
Right, I think Spurs are out of our reach, but as for the other 3.
Newcastle - Win our game in hand, and beat Newcastle, and we've got them. So we don't actually need them to mess up, it's in our hands.
Chelsea - Win our game in hand, and beat them, and we're 1 point behind. They will drop a point between now and the end of the season (in a game other than their one against us), and indeed, many more points to boot.
Arsenal - Win our game in hand, and we're 7 points behind. Obviously that's a few, but it would be pretty phenomenal if Arsenal managed to not drop more than 7 points between now and the end of the season. They would have to play a great deal better than they are to do that.
So yeah, I don't consider there to be a problem with the other clubs messing up, because they all are pretty much guaranteed to. The big problem is, as you then said, us. At the moment, we're not going to get the wins together to do this. Even if we were currently the team in 4th place, you'd still back us to drop it to these other clubs because our form is worse. However, whilst the problem is us, this at least means one good thing. We (the football club) can do something about it. It is still in our hands. Rafa's leftovers did a pretty good job at the end of last season. These guys should be able to give it a better go. Remember, all 3 of the clubs that we need to overcome are in pretty bad form themselves.