Hope this helps somebody!
I had this same issue in Los Angeles County (Long Beach) California. Long story short: seems like a hardware issue that was fixed by getting a final replacement for iphone 5.
This issue only started after I got my 1st phone replaced for an unrelated issue, so I was already suspecting a hardware issue.
The Problems
On the new phone there was reception issues when LTE was turned on. It would show 2-3 bars, then 1, then Searching, No Service, Activation required. This kept cycling.. maybe once every hour. Turning LTE off helped make this issue better and data was pretty fast temporarily. Later on this cylcling started happening even with LTE turned off.
This also caused the battery to drain about 20% overnight on standby.
Attempts to fix it that DID NOT work:
reseat SIM card
reset all phone settings
reset network settings
power off/on
DFU restore, and then NO itunes restore from backup. then reset network settings, reset phone settings.
DFU restore, then restored from my own backup. then reset network settings, reset phone settings.
swapped sim card, reset network settings, power on off.
Updated IMEI for new phone with AT&T, reset network settings, powered off/on
What DID work:
After being able to reproduce the issue with a tech at Apple, he gave me a replacement. The issue was fixed immediately without any restore/reset needed. Definitely a hardware issue. LTE works perfectly now and properly switches to "4g" when LTE signal is not there instead of "no service".
This is after having swapped out my phone for out for out-of-the-box cosmetic damage 4 times. Volume/sleep/power buttons are loose on ALL of the phones I saw, to varying degrees. Quality control on these new phones are very frustrating.