I think that a certain amount of competitiveness is a good thing. It can drive people to do more, to be better. This is true with games, exercise (yes exercising can be -very- competitive), or any hobby you might have. Heck, even writing fan fictions and drawing fan art can feel competitive with the right friend.
If someone is so competitive that they have to have the best character and they feel like it ruins their game if they loose then there is a problem. I have a lot of friends like this. They are good people, but I can't stand to play games or the like with them. If it's a pen and paper rpg then they would create a character in such a way that they would basically take the glory. That's not fun for other people, though it may be fun for them.
Now, for my little competitive anecdote. My husband and I were doing a work out for P90X. It was the side-tri-rise
move. Now, for this one we were only suppose to do 25 for each side. Well, my husband and I were facing each other and out eyes locked for a moment. There was an unspoken challenge there. We had to see who could do the most. We ended up doing 40 each side and collapsing in exhaustion. That is one painful workout if you do it right.