I had the same problem with our iPhone 6+. I tried everything in the thread: shut off/turn on, hard reset, remove SIM card/replace/hard reset, Airplane mode on/off, reset network settings, etc. I just got off of the phone with AT&T and our problem has been fixed.
My husband broke his phone and switched service to an old iPhone 4 while we waited for insurance to send a replacement. The 4 was only connected and used for 2-3 days. When we connected the 4, we had to go to AT&T for a new SIM card, then another new SIM was sent with the new 6+. We connected the 6+ and one day later had no cellular connectivity. After trying resets on my own, I called AT&T to see if they could send a reset. In our case, when we activated the new 6+, the ICCID was not done correctly. Something about downgrading to the 4 then back to the 6+...
Just some info for anyone having similar issue.