Balls In Boxes

New puzzle o'clock! I should probably be working instead of blogging, but blogging is so much more fun. Anyway, I found something of a neat puzzle on the Putnam 2010 exam. Usually those exams are just filled with semi-impossible math problems involving stupid functions, but there was one question on that exam I thought was pretty cool as a puzzle:

There are 1000 boxes, labelled 1 through 1000. There are 1000*N balls in the boxes, distributed randomly between the boxes. You will be playing a 1 player game, trying to average out the balls in the boxes, so that each box has exactly N balls in it.

The only legal move in the game is to select a number K and move exactly K balls from box K into any other box. So you could go to box 247 and move exactly 247 balls from that box into any other box. If box 247 has fewer than 247 balls in it, you cannot do this, so you don't get to move any of them. You are allowed to keep making legal moves as long as you like, and you win if every box has exactly N balls in it.

Given that in total there are 1000*N balls in the 1000 boxes, for what values of N is it always possible to win, regardless of the initial configuration of the balls?

Naturally, the fundamental nature of the solution is not particularly sensitive to the number 1000, in the actual Putnam they used 2010, cause thats how they roll.