Thanks for all of your thoughts. I will try the firmware updates, that's a good thought.
But I do think it might be the speedlight, although not firing the camera does seem to be a strange way for that to manifest. It seems to be worse in the last few shoots, and this time of year I am shooting inside more. Also, I got the SB 900 about two years ago, and that coincides with about when the problem started. The SB 800 never stopped firing so that makes sense. What do you do about the flash overheating other than slow the shooting? Would it be fixed by changing batteries? Or is it an indication the batteries need to be replaced? Or is there nothing you can do but pause?
I did consider the camera batteries as an issue, but the D800 batteries are all brand new. Also, some of my memory cards are old (although I've never had a problem) but I had to get some new larger memory cards when I got the D800 as well, so those probably aren't the issue either.