I had this same issue late in 2013, but resolved it and have had no issues since.
This is *not* inherent to Windows Phone or Lumina. It is inherent to MMS/SMS messaging. I am a long time Windows Phone 7 and 8 user. Recently after changing to the Lumina 1020, I had similar issues. Group texts not working, any text appeared
to have "media" attached, or if media was attached couldn't receive it. No error code. I could send messages with pictures but couldn't receive them. After many phone calls to ATT, going to the store and texting with technicians, troubleshooting myself,
I finally talked to a technical resource at level 3 support who shared something really helpful with me.
I am a hoarder and keep text messages and media in the place they came in. So if you are like me, read on...
SMS/MMS based messaging is about the lowest form of communication, invented exactly 30 years ago in 1984. It does not have the same control over individual messages that other services have. It is FIFO (first in first out). So if 1 group message with
pictures gets stuck, then all the subsequent text messages with media or group message (which use MMS) also start to fail. MMS/SMS continues to try to send the bad one over and over again and the rest of the messages are just standing in line single file.
So agreed if this is a new phone issue, the imessage response above for former iPhone users has helped me in some cases. Or making sure MMS is turned on is required for this activity on ATT. But if this is an existing phone, worked before and just starts
happening one day, go back and look for a group message with picture/media attachment, delete that one (or in my case several) that were potential culprits, and test with a friend media only first and then group, group with media.
Side note : before you delete a pic if you want to safe it, click on the picture in the message string, bring it up and hit the ... to save to your phone so you don't lose it. Just get it out of the text thread.