I got a new iPhone 5 last night and happily came home to try to do a backup and restore from my old iPhone 4 and hit this problem with trying to activate iMessage and Facetime to no avail. I searched the net and came to this thread. Followed the instructions given here and just could not get it to work. I knew it is not my Apple ID because I can still disable and enable it on my iPhone 4. Finally, I did the following to make it work:
1. Backup the iPhone 5 (once you sync, it is backup already).
2. Reset and wipe out everything on the iPhone 5. That is, tart from scratch.
3. Once I completed the set up process, I immediately login to my Apple ID and then start to enable and activate iMessage and Facetime WITHOUT doing the restore first. Well, obviously, it worked!
4. Now that iMessage and Facetime are both enabled and activated, I then plug the iPhone 5 into my MacBook Air to start the restore. After the Restore, everything is still working - iMessage and Facetime are both still enabled and activated and working like a charm.
If you have tried everything else and it did not work, you can try this to see whether it works for you. It does not take that long to go through the process.