Thanks Lawrence, that's exactly what I would have thought, that a regular MMS message should come through without cellular or wifi, that's why they sell message packages aside from the data package. But that is not working on my phone! The one that happened this morning didn't even come through with wifi ON!!
The message was a group MMS sent from my sister who has some kind of android phone so iMessage is not even an option. It was sent to my other two sisters and me. One of those sisters has an iPhone like me and the other one is still using an old Razr with a T-Mobile acct. Although I don't see how what the other recipients are using should matter, just thought I would include that.
Another person on here mentioned something about when the phone is in sleep mode it will revert to cellular and I think that is the key here. I'm thinking my phone was probably in sleep mode at the time the message was sent, however, I used it on and off (with a good wifi connection) for a couple of hours after that before getting ready to leave and turning on cellular and this is when the original message, along with replies from both of the other recipients, finally came through! The message should have come through without wifi or cellular yet it wouldn't even come through with wifi!
This has also happened before and it really gripes me because Apple and AT&T claim that it should be working. Lastly, even though this doesn't apply here since the sender isn't as iOS user, I kinda thought that setting about "send as SMS if iMessage is not available" only meant for the sender's phone. In other words, if the sender had cellular turned off and no wifi connection that it would send the message as SMS.
Thanks again for your input!
Dave