I realized, that I was being sent an mms message when I get the dreaded "download" button from the sender sending me the message, and after a few moments I noticed that my Mobile Data ,function was off, and as soon as I tapped it on, whola! The message came through. I am quite avid with manually turning off my data when I'm home, and then turning on my wifi. And forget to turn on my data when I leave my house. I also am running 'juice defender' in which case may automatically tone down my data function, then periodically it turns on momentarily to bring in info that requires data.
I've also had to uncheck the auto switch feature in my note 4 because it's designed to look at the strongest data signal between wifi and data.
I said all this to help others, look at their settings a little closer and go through some trial and error situations to determine, 1. If your mobile data is off, 2. See if a program may be controlling your mobile data, 3. See if you have the auto switch feature enabled.
As far as a Verizon user goes, this send to have a definite factor as far as sending and/or receiving MMS messages.
***the auto switch feature can be found, by going into settings, then wifi, tap on the 3 vertical dots at the top, then tap on advanced, then go to auto network switch. If there's a check in the box, this may be your problem if connected to wifi, it may not be, but it's worth a shot to see if that's your problem or not.
Sent from The One and Only, Amazing---> Note 4