I'm having this issue as well, so I thought I'd add to the thread in the hopes of helping to discover the common denominator(s).
I have an iphone 4 (an early one that I got in the first month it came out) carrier is AT&T and i always have less then 5 gigs free but it was never an issue before. started having issues several weeks ago while still running 6.1.2. Just updated to 6.1.3 and no change. I looked at the "apple systems status page" and facetime was not having issues. I have tried on two different wifi networks (both home networks and secured with a passkey) and got the same thing on both. I make (or accept) an FT call, and then it says "connecting" and then it says connection lost. I have not tried the factory reset because enough other people said that it did not fix the problem and it seems like a real pain in the neck. I tried turning off bluetooth, removing all icloud syncing and/or backup except findmyiphone, i tried removing my passcode lock, checking restrictions (they were off), checking date/time (it was set to auto already), deselecting the additional emails to be contacted through in facetime settings, turning facetime off and back on, doing a hard reset on the phone, killing all apps running in the background, and I can't remember what else before I tried calling 1-800-apple. actually got through to someone fairly quickly who told me that I should go to the facetime settings and log out of my apple account there and do a hard reset and log back in. tried that and no change. he said if that didn't work that I should back up my phone via the normal method on itunes and do a factory reset and then sync it. and if that didn't work that I should manually back up each segment of my phone on icloud or elswhere (for instance pics on harddrive) then do another factory reset, and then manually put everything back on my phone without using itunes. I tried logging out of facetime and back in and no change. have not tried the factory restores yet cuz I'm not at home. I am at my dad's house and used his ipad to facetime my sisters iphone so i know it's not the wifi. I tried turning off wifi but apparently I need a 4S to do facetime without wifi. apps that I think I may have added since the last time that my facetime worked - belly, google maps, gardenguide, gardenID, dr. vs virus, glow hockey, itunesU, Podcasts, filesanywhere, couchsurfing, pinterest, printnshare, 2do, and so on. deleted several of them and no change. when I get home, I guess I will have to try the factory reset and see if it works with nothing on it and for how long and start adding things back. this is ridiculous. I get the feeling that maybe i'd be better off just giving up and waiting for my sister to upgrade to the 5 and then give me her 4S. anyway, if anyone figures this out and can alert me to a solution that works I hope they post a link to their amazon wish list and i will send them a present cuz I sooooooooo don't want to spend anymore time on it.