My kids have two nationalities. If they will be competing in any sport on a national level, can they choose to participate in the competition of both countries of which they are a citizen?
While I agree with Gaël Laurans that this will mostly be a matter of the respective international sport federations, the choice does not necessarily have to be for life. As a specific example, there is an ice hockey player who represented Canada on the 1994 Winter Olympics and the Czech Republic on this year's Olympics. This might still, however, be possible only in some disciplines.
As far as I know, such rules are set by international sport federations so they differ from sport to sport. I don't know many details but it's often impossible to play for several national teams or selections in your career. Once you went to a world cup or olympic games or were selected for an official game, you are typically committed for life.
However, since this is regulated at the federation level, the rules aren't even unified for the olympic games. For example, @Ondřej Černotík mentioned a player who played for two countries in ice hockey but I believe this is not allowed by the FIS (the ski federation).
Furthermore, in some (most? all?) sports, you don't have to choose before you are ready to play at the most senior level. It's not uncommon for people to play for one country in “junior” competitions and play for another country later on.
Some sports (rugby) even allow players who don't have the nationality of the country in question to play for the national team under some circumstances. Still, once you played for one country, this becomes your “rugby nationality” and you can't play for another representative team, ever.