It depends entirely on the way you broke up, but for the most part it may be difficult for one or both parties involved.
If the breakup was mutual then a friendship may actually be a possibility, but this is rare. As time passes and new relationships are formed it is possible that a friendship may be rekindled if past feelings are laid to rest. If the new persons involved however will want that is an entirely different story.
How To Tell
Friendship-possibilities can be calculated by how deep and long your relationship was. The longer and more invested one was (emotionally as well as sexually) the harder it will be to remain friends afterwards. If the romance was fleeting and not overly physical or emotional and both parties mutually agreed that romance was not working, then it is a good chance they may remain close friends.
But What About Those Odd Couples That Are Still Friends?
While there are exceptions to the rule, it isn't always the case or the norm. In all relationship breakups someone was hurt in one way or another, and this pain does not easily heal. Staying close friends may be lemon to that wound.
Let's Just Be Friends...
One party may be willing to return a relationship to a platonic state and would be able to keep it that way, but the other may not. However, the hurt party may keep up appearances and agree to a 'friends only' relationship in the hopes of a future rekindle. These 'friendships' may become awkward eventually.
What About Staying Friendly?
Since it is nigh impossible to completely avoid past relationships, a certain social friendliness can be maintained. How each individual party deals with these is up to them and many would say it is the 'mature' thing to do. Be it 'mature' or 'malicious,' one should tread carefully around exes. After all, they may be friends or acquaintances of your next big romance...
How To Avoid The Situation All Together
When beginning relationships it would be wise to think things through beforehand -- at the very least try to keep a hold on the depth of the relationship: don't jump too quickly to conclusions, don't be overly eager to give away your heart and take it easy on the physical aspects.