I had the same problem when using my LG G2. But, I sometimes was not able to receive MMS's from Android AND iPhone users. It got very frustrating. I am always in full, or near full LTE, or on a stable WiFi connection. The only, temporary,
solution was to use another messaging app. I tried a handful, got sick of that, so just used Hangouts as my default. That fixed the problem!... For a couple of weeks, then I would get the same "broken" image thumbnail where the picture or text that the MMS was supposed to be.
Now that I'm on my G3, I noticed this problem was even more serious. Still in the AT&T store, I received an MMS, and was not able to download it. They exchanged it with another G3. Got home, same problem. It's a complete hit or miss, being sent to me from and Android or iPhone, I receive about 80% of the incoming MMS's, and the rest I will ask the person to send it to my email, or use whatsapp if that's an available option.
I have called and emailed about the problem and I've only gotten the responses that you think a "technical level 2" employee would give.