I recently bought a Galaxy Nexus, and I'm having problems receiving MMS messages. Every time someone sends me an MMS message, my phone seems to not be able to download the message.
I had the same problem and fixed it by going into:
settings, wireless & networks "more", mobile networks, then you need to "enable" data access over Mobile networks
It will then have a check mark on "data enabled"
After I did that I immediately was able to send and receive MMS.
Thanks for the help!