We have three iPhone 4's and one iPad 2 in our household. We never had much reason to use FaceTime on our devices since we all resided in the same house, but we tested this new feature when it first became available during one of the past iOS updates.
At that time, FaceTime worked as expected. Since that time, one of our family members has moved to a different location and we thought now would be a good time to use this feature.
We soon discovered that we could not make a FT connection between any of our devices in any combination. As previously reported, each of the devices could make contact with any other device, but FT would stall at the "Connecting" prompt until it finally timed out and failed a few minutes later. This would happen whether the devices were on the same wireless network at either house or if the devices were at different houses.
So after much research, including this thread, I decided to begin the repair process by restoring my iPhone as "new" and not restoring any back ups.
Since I use Google Sync for my email as well as iCloud and my iTunes on my PC for backups, I did not have to worry about my emails, calendars, notes and such. I used whatever means that any apps that I was using that created data to back up or export that data so that the data could be restored to those apps after the repair process was completed.
There was nothing of any importance in any of my text threads, so that was not an issue for me. I was not concerned about losing any call records and I keep my voicemail box clear of old messages.
I started by making note of the apps that I intended to reinstall and then I deleted every single app, song and video from my iPhone, leaving nothing but the bare iOS system installed. I then took note of the amount of space being used when everything was gone.
Once I was sure I had everything exported and notes were taken on all of the settings I would have to restore, such as e-mail accounts, wi-fi info and such, I attached my iPhone to my iTunes and performed a restore as new.
During this time, we discovered that the iPad was able to connect to a non-family member in another state via FT. This was good news since I would not have to perform the restore procedure on that device after all and that would also give me a device to test my iPhone against.
It now appeared that our problems were only with our iPhones and we determined that this was because the three iPhones had their iOS updated though the same iTunes account on my PC while the iPad was updated to that same iOS update as a stand-alone device over the wi-fi in our house. The three iPhones therefore got a bad iOS update that they shared, but the iPad got it's own iOS update, which was good. This probably explains why not everyone out there is suffering from this problem. Some of us updated our devices from a bad iOS update, while others downloaded a good one and did not corrupt their FT on their device.
As soon as I finished restoring my iPhone, I set up my wi-fi connection and configured FaceTime, Messages and iCloud. I then placed a successful FT connection to our iPad. Once that was done, I reattached to iTunes and reinstalled only the apps I intended to use going forward, along with my song and video collections. I also imported (not through and iTunes or iCloud back ups) any data in any of my apps that needed it. I then set up my Google Sync and other e-mail accounts and allowed everything to download to the iPhone. At no point was any restore from back up (iTunes or iCloud) allowed.
After all was said and done, all of my apps and settings were back like I needed them to be and further tests with FT showed it to be working as expected. I finished the process by performing a forced back up of my iPhone in both iTunes and iCloud to overwrite any bad backup that existed.
The only thing I had lost was my text threads, which was not of any concern to me.
As a side benefit, when I performed the restore I had gained 5GB of usable space on my phone. I am not sure where all of that space was being used before, but I am happy to have it back,
I now have successfully applied the iOS 7.02 update to my iPhone and I can report that FT still works as expected.
I now plan to perform this procedure on the remaining two iPhones and finally put this problem behind us.