Not receiving text messages

Two-ish years ago I noticed my phone, an HTC mini, was not send and receiving all of its text message, both SMS and MMS. I called and spoke with a rep who had me do a number of things - check various numbers, increase stored amount of texts, clear cache. Per the rep we tried everything short of a factory reset. It fixed the problem for a little while, but after a few months I started not sending and receiving some messages again.

This past Dec I got a new SIM card and when that didn't fix the problem I ended up purchasing a new phone, a Moterola G3. The new phone came with a new SIM card and I am again experiencing this problem. The problem seems to be one linked to ATT's ability to send and receiveing packets (are texts sent as packets??) or my account - as the problem exsited through 2 SIM card changes and a new phone.

It isn't all the time, or with a particular schenario. Sometimes it is a regular one on one text, most often it is a group text, and I rarely get pictures that people send me, especially if the picture is in a group text. I'm not sure what else to try as a brand new phone didn't even solve the problem, but I'd really like to have all of my texts come and go correctly. 

Thank you