My husband and I both purchased Samsung S5s about 2 weeks ago and have both been encountering the same problem: If wi-fi is turned on, we cannot receive MMS messages. We know that mobile data has to be turned on to actually *download* the message, but the MMS notification does not even come through until wi-fi is turned off for a few minutes.
I have seen other people with this problem, but no answer other than "just turn off the wi-fi" which, to me, is not an acceptable answer. Our wi-fi is almost always connected at home and especially at work where both of us are in buildings where the 4g/3g signal is very weak.