For almost two months now I have been experiencing this issue on my iPhone 6 using AT&T. It happened under iOS 9 and it's still happening under iOS 10. I had no issues until I had my 6 replaced under AppleCare for a bad battery and since getting the new one I constantly get "No Service," or my phone switches to 3G. In most cases the phone will not switch back to LTE without me putting it in airplane mode and then back again. It's frustrating and one of the main reasons I'd planned a switch to Verizon, assuming it was carrier related. Now I'm wondering if that's the case. I think it's time to dust off my old HTC phone and see if the service is the same or if it's indeed an iPhone issue.
Either way, I'm holding off switching carriers or upgrading to iPhone 7 until I hear people's issues are getting resolved somehow. 🙁