After six weeks of reading posts and dealing with this myself, I can tell you that it's something to do with the AppleIDs and Apple's servers or verifications. Or permissions.
This is my take on it.
People having this problem have some things in common
- Almost all have iPad Mini 3. A few have iPad Air 2
- They have iOS 8.2 or 8.3
- They have an AppleID that was never used for FaceTime or iMessage before about the middle of March. The AppleID might have been used, but not for iMessage or FaceTime
WIthin in that there are some puzzling inconsistencies. Some people can use FaceTime and iMessage on other devices but not the iPad Mini 3, other people can't use their AppleID for FaceTime and iMessage on any devices. And people who have old AppleIDs can use them on everything.
People who can use the AppleID on other devices for FaceTime and iMessage might have a different problem than those who can't use the AppleID anywhere else for FaceTime and iMessage.
My two cents. And for the record, I don't know why mine started working again but the engineers did have my account in test mode.