If you don't want to downgrade back to iOS4, here's a "solution" that has been working for me. I've been experiencing this problem for several weeks. It first started with iOS5 beta 6 for me.
- Purchase and install Phone View
- Plug your phone in and let phone view backup all your texts.
- Delete SMS threads. I went through my SMS mailbox and deleted all old threads that hadn't been updated since January of 2011.
- Quit messages (as in really quit, not just put in the background)
- Power off, power on... messages come in.
I've had to repeat the steps of deleting and powering off a number of times as the mailbox full warning has returned every couple of days for weeks. I have now deleted all threads that hadn't updated since June 2011. According to phone view, I have around 50k messages. That seems to be the limit.
Of course, this is not the best solution as I'd have preferred to have kept all of my texts on my phone, but at least they're backed up and accessible on my computer. And for me, it's certainly better than restoring as a new phone and losing all of them.
I was ok dealing with this issue while iOS 5 was in beta... it was beta, after all. But since this issue has been present since beta, it means they released the OS without addressing this known issue. That's just poor business and it bugs the **** out of me.
...I, like all of you, am hoping that Apple will address this swiftly. If anyone else has a better solution than this, please post it.