I have had the same problem since mid July on an iPhone 4 with vodafone in the UK. Literally, with no changes made that I can recall, facetime stopped working over night. Just keeps saying connection failed.
I have tried all the various resets, been to an apple store and spent hours on apple telephone support. None of which have worked.
I have tried other people's wifi which didn't work.
I tried scrubbing the phone to new and restoring from a backup via iTunes. All this did was reinstall the problem.
It can't be an issue with my apple ID as FaceTime works with it on my iPad.
I scrubbed the phone and without restoring from an iTunes backup or icloud I signed back in to icloud to merge my contacts and such like. After setting the phone up I tried facetime and it worked.
Next I tried syncing my apps and resyncing my photos. After this facetime stopped working again with the same connection failed message.
I then deleted my photos again and still facetime didn't work.
After all this I have come to the conclusion that the problem is App or App & iOS related. I have too many Apps for me to one by one reinstall and try facetime. I would be there for days doing this.
It's a really really frustrating problem.