HT4759: iCloud: System requirementsLearn about iCloud: System requirements
Currently Being ModeratedMar 29, 2013 6:22 PM (in response to Banks123456)
This happened to me as well. **Warning: The following steps will remove all network passwords from your phone and you will have to re-input them. Unfortunately, this is unavoidable**
To resolve, I went to Settings > General > Reset > Reset Network Settings and accept the Reset Network Settings option.
Afterwards, I was able to log into the Game Center without a problem.