It could also be that you have somehow lost your MMS carrier provisioning. As you may or may not know, text messages (SMS) are exactly that - text only messages. Once pictures, music or video is added to a text message, it is no longer a text message, it becomes a Multimedia Messaging Service message (MMS).
SMS uses the carrier cell network but MMS uses data. It doesn't use your data bucket like email/browser/etc., but it is a special data bucket for MMS messages that the carrier has to provision you and your SIM for. Often, especially when changing devices or making changes to your plan, the provisioning gets "lost" and only a call or visit to the carrier can fix it. So as Meg mentioned, if the MMS gateway determines that you don't have MMS provisioning, it will send you a link to download the picture/video/music that was sent to you (which you can then download using "non-MMS data"*). I recall from the 'old days' also that sometimes there was a day or two delay between the original message being sent until you go the link -- especially if the sender and receiver where use different carriers. Remember also that many people do not differentiate between true text messages (SMS) and "text messages with pictures" (MMS).
* usually, cell service consists of: cellular network calls, text messaging (SMS), picture messaging (MMS), data (for emails/browsing/downloading) and voicemail.
This doesn't explain everything applicable to your situation as you've explained it but I am mentioning it as a possibility; and it might apply to others reading this thread with a similar problem. Good luck and let us know how it turns out.
