Still happening here on my iPod 5th gen. - I keep having to sign back into Game Center which flakes out sometimes dozens of times a day (more often the more different games I play).
Worse it often gives the moronic "cannot connect to Game Center" error, which of course is rubbish. Because if I switch to the Game Center app itself I can connect mere seconds later.
And sometimes I find I also need to swipe upwards to force quit both the previous game and Game Center's app, before login with work, but still about five seconds later (as long as it takes to force quit both) I can connect, so this is not a connectivity issue. It is a crappy lack of testing before release issue.
I swear I am at my wit's end after a couple years of this !@#$. For about six months the issue seemed to go away, I think when iOS 8 first came out. But then it came back a few months ago *groans* All told I think I've had a total of about a year of this happening over non-consecutive periods since I bought my iPod for gaming back in 2013.
Phoning AppleCare is a waste of time. They have zero clue about this issue, and I have been through the usual troubleshooting steps a dozen times trying to solve this and iCloud issues that are thankfully gone... for now (iCloud services kept being disabled randomly by iOS back in iOS 7), to no avail. Not that AppleCare reps aren't always polite and trying to be helpful, IME. But the fact is they have yet to actually be helpful, no matter their cheery and earnest intent.
In the end I feel I am at the mercy of Apple engineers finally realising there is a problem and giving a **** about fixing it. Luckily that seemed to happen with iCloud (not that AppleCare are aware). Perhaps the six months a while ago when Game Center stayed signed in was a fix that fell by the wayside in a later patch, or was just randomly and inadvertently fixed, only to be broken again later. I don't know. I only know this: I am ******.