After doing some research, I found out how to stop this from occurring; the steps are below. I've already had to apply this "fix" at least three times over the past three months, so I know it works, but I'm just not sure why it happens in the first place.
Here are the steps to correct the issue:
- On your phone, go into Settings
- Swipe right to the "applications" tab, and select "messaging"
- Turn off Group Text by unselecting the option
- Restart your Lumia 1520 by holding the lock button - camera button - and volume down button at the same time until your phone vibrates and restarts.
- Once your phone restarts, go back into Settings and turn Group Text back on.
- I then delete any group text threads where my phone number appears as a recipient.
Again, not sure why this happens, but it started happening in October 2014 and has occurred about four times since the time of writing this post. I am part of the Developer Preview program so it may have something to do with that, but hopefully it is fixed before the next official OS release / update.
I'm running Windows Phone - 8.10.14219.341, with Lumia Cyan. Anyway, hope this helps someone else.