I had the exact same problem you describe above (Galaxy Nexus on t-mobile USA) it is because mobile data is turned off.
In Jelly Bean it is: Settings > Data Usage > mobile data
Note that I have to have mobile data turned on PRIOR to sending an MMS OR receiving one. If I receive an MMS with mobile data turned off, I will get the notification of a new message and I will receive the message with a download button. But if I do not have mobile data on prior, the incoming MMS attachment will not be received. Even if I turn it on after the message was received.
For some reason when your phone provider enables you with the ability to send and receive MMS you must have the Mobile Data enabled, even if you are using Wifi, if the Mobile Data is enabled you will be able to receive and send MMS, even if Wifi is showing as your internet on your device.
It is a real pain, as if you do not have it on, the message can hang a lot, even when turning on Mobile Data, and might require a reboot of the device.
~ Answered on 2013-02-14 13:23:38