I have the same exact problem. I open game center, sign in if asked, launch my game, play it and after a random period of time the game crashes, I reopen it and then get a gamecenter login prompt inside the game, I enter the password and it can't connect. I close the game and go back to step 1.... it's been happening for a few months.
Also If i charge my ipad with wifi on, sometimes icloud will try to do an auto-backup, this never ever works because it always needs me to enter my password first. But if I manually tell the ipad to backup to icloud then i'm never prompted to enter the password.
I tried turning on 'find my ipad' as mentioned as a posssible solution to the gamecenter issue in another thread, this didn't help, and a few days later I noticed 'find my ipad' and icloud backup settings all turned off.
so does Apple read these posts? I have an ipad mini non-retina, wifi, very good wifi connection with no interruptions ever, and a reliable fast internet connection that has no problems.
The problem is at Apple's end... when will it be fixed?