This message was posted on the wrong thread. I offer it hear in case it might help somone having problems with FaceTime.
There are so many suggestions offered to solve the FaceTime problem that I thought I would add my current experience. I took my iMac to the apple store and they found no solution. Their last effort was to reinstall Mountain Lion (ML). It didn't solve the problem. I opened a case with apple and worked with them for about two hours on the phone to explore possible solutions. Finally a system dump of the internal data was taken after an attempt to sign on to FaceTime. The data was sent to Apple and a team analyzed it. They discovered that my computer's logic board had not been serialized. Back to the Apple store where they serialized the logic board. FaceTime now works. This may not be a solution for you but it might lead to other ideas. Merry Christmas.