Here's what worked for me...
First, let me say. For shame Steve-Job-less apple for implementing such a ridiculous solution. One leg up I've always had on other phones is that I know the history of all text message exchanges with anyone I've ever texted. And while that's slightly packrat-ish, with the declining costs of storage AND the opportunity to sell it at a cost (I've always bought the BIGGEST of any of each of your phones), it should remain an option. I hope you re-visit this constraint. OK...off the soapbox.
Yaayy, me for registering to post a solution that worked for me.
It seems that the speculation of the maximum number of text messages is right. Just how many is still TBD. It's not 75k because I got well below that and was still not getting texting. And how do I know how many I had? Introducing PHONE VIEW. (I got this solution of the apple community forum but I started here so I wanted to share it here.) Phoneview is a cool little tool that you can use to view and archive (not edit or delete) texts, pictures, voicemails, files and most other auxiliary memory features.
So here's the solution that worked 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 old threads until the new text messages started blinking in. I've seen published numbers of the limit being 75k and 50k. I had to get even lower. It kinda doesn't matter since you now have archives.
Quit messages (as in really quit, not just put in the background)
Power off, power on... messages come in.
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.
Make this right Apple.