Same issue for me on my Lumia Icon - just received Denim update and now I can't get or send MMS. Do not have the ability to set anything in cellular+sim other than the internet APN.
I too have been experience the same situation on my HTC 8X. I had this issue before installing the 8.1 Developer Preview, thinking the preview may fix it. I too am thinking it's a hardware issue as it keeps dropping the cellular data connection.