If you’ve cancelled game center sign in enough times, the OS disables game center for that game. Prior to iOS7, you could manually sign in again using the game center app, and when you launch your game again it would sign in.
However, in iOS7, it appears that when the OS disables game center for your game, it’s completely disabled. I’ve found that using Settings > General > Reset > Reset All Settings will restore game center functionality for games that have been disabled in this manner.