This just happened to my iPhone 5 right AFTER I installed the latest software update. Oddly enough, my iPad 3 is having the same issue, but there was no update for it when I checked. However, my husband's iPhone 4 AND his iPad 3 are working just fine. I have AT&T service, which originated in FL and am now in northeastern NV.
None of the troubleshooting suggestions have worked. Also, my iPad is on a different data plan from the phone - they're just linked through iCloud.
I haven't called Apple Care yet. Based on everything I've read, sending the phone in and not having it for nearly two weeks only to have the problem recur is unacceptable. I live in an EXTREMELY remote area where the nearest ANYTHING in any direction is 4 hours away, so not only do I NEED the phone for safety and/or emergency help, I will be FORCED to mail it in for repair/replacement. Gggggrrrreeeeaaatttt....
If anyone finds a legitimate solution from Apple or their service provider, PLEASE share it here. Thank you.