Suzanne Collins never gave us a specific time of year that the annual games took place, but compiling several facts you can conclude that it happened during the summer.
This would make the most sense because of when the kids would be out of school. The kids would be on odds if some kid was fighting to the death. Not to mention you would not have to interrupt school to watch the games.
The most confirming fact is that the Victory Tour was halfway between the games and it was in the winter. Logically you can conclude that the games occurred during the summer. (Talk about a scary summer vacation!)
