iPhone 5s SMS message problem

I have had a big problem receiving group text messages on my 5s.  My main group is a combo of apple and android, and of Verizon and AT&T,  They can see what I send but my messages bog down coming back.  I don't get them sometimes for 36 hours and then they come randomly, sometimes dozens at a time.  Alternately, I get dozens of  "Message Not Found" messages at once. 

The Apple Store tried a full reset of the phone.  The AT&T Store replaced my SIM card.  The Apple Store even replaced my phone when neither of those fixes worked.  The new phone worked for about 2 days before the same thing started happening again.

The only other thing that I can add is that in comparing messages with the others in the group it seems like a picture message may have started the bogging down but I can't be certain.  It has happened in multiple groups, but especially this main group.  It doesn't see to affect iMessages, or individual texts.

Any help with this is greatly appreciated.  Having reset the software and replaced the hardware, I am almost certain that this has little to do with the phone itself.  Thanks in advance!!

P.S.:  This isn't an issue of signal strength...I have a strong cell signal and wifi signal in my home.  Often this will start randomly while I'm texting in a thread and receiving messages fine.