In the 2013/14 season Sunderland were bottom with 26 points after 33 games.
They finished the season 14th with 38 points. As per the BBC article posted above, they had a pretty tough run-in too (Man City, Chelsea & Man Utd all away).
Leicester won SEVEN of their remaining 9 league games to stay up in the 2014/15 season. After 29 games they were on 19 points and 7 points from safety. Finished the season on 41 points. Not saying we'll replicate this but football's a funny ol' game.