iPhone 5S, LTE does not work I have an iPhone 5S 64 GB, ME312LL/A) I purchased as a refurb from a reputable vendor in November 2014. It's my wife's, and her previous phone didn't have LTE, so neither of us noticed right away that even in areas AT&amp;T has good LTE service, her phone never registers an LTE connection. Only 4G.

Update:

When a SIM card from a phone with working LTE is inserted into this iPhone 5S, the service indicator shows LTE. I suppose I need to contact AT&amp;T then?

By all appearances the phone had never been used (original packaging, pristine accessories,OEM screen protector wrap still on). What would cause a phone to function normally all ways, but not register an LTE connection? 

I've tried:


toggling LTE off/on in Cellular Settings. 
checking for carrier updates
resetting network settings
I assume you should try to test the SIM with another phone in order to see if it really works.

Test

All you have to do is pretty simple:


Get your SIM ejector and take it off from your current iPhone
Get another iPhone and put it in
Unlock the SIM on the new iPhone and make sure LTE is activated properly under Settings > Cellular > Cellular Data Options


Once done this, I'd definitely try to contact your carrier's support (which if I'm not wrong is AT&amp;T in your case) and report them the result of the test and see how they can help you.

Hope this helped you out, and don't forget to let me up to date on the question!