If you're unable to login to FaceTime and iMessage because the login process repeatedly loops, check the time settings. If the time is not set to automatically adjust (Settings > General > Date & Time > Set Automatically > On), you might find that the login sequence for both FaceTime and iMessage loops. Once you set the time to adjust automatically, all is back to normal.
[kirkmc adds: I've never had any problems, since I've always had that setting set to On. But I've found that a number of odd settings can affect logging into Apple services, so it's worth pointing out.]
Mac OS X Hints
http://hints.macworld.com/article.php?story=20130530174003397