Cannot sign in to iMessage and Facetime with Apple ID (only one or the other)

Hi

I have an iPhone 5 running the latest iOS (on AT&T), not jailbroken.

I have recently got family to use Facetime and have been using it more and have noticed I am unable to use my Apple ID for both Facetime and iMessage- if I am signed in to one of them, I am unable to sign in to the other- I get in a loop of the phone repeatedly popping up a password entry screen for the Apple ID.

If I cancel the process and sign out of either, I can successfully sign in to the other. But no matter which I sign into first, I cannot sign in to both...

Anyone else had this problem? Anyone solved it?