Circling back so everyone knows what the problem was:
Problem solved. Here's what I found:
Last night I took the very same 100 9mm cases that I had problems with earlier, and ran them through a single stage press to size them and re-prime them with new CCI primers, using the primer arm on a Redding Ultra-Mag press. This is a very sturdy press and all primers were easily seated fully and perfectly straight in the pockets. It's a tight press - indeed, the primer arm has never been used. I took these rounds to the range today and they worked 100%. Everything is right with the world.
The problem: it appears that for some reason, the priming tool that I was using did not seat the primers either fully, or straight - not sure which. What I find very interesting is... I also re-seated primers in 100 cases by running them back through this same tool and loading them. These primers had originally been seated with this tool. At the range today, I had 6 failure-to-fire cartridges in one 15-round magazine. This tool is an early (1960s?) RCBS bench-mounted priming tool that loads one primer at a time. It has now been retired.
Now, my only hassle is the fact that I have ~2,000 cases that have been primed on this tool. I will punch those primers and re-seat new ones. At least I did not load them up without testing! This problem cost me about 2 bricks of primers and a bunch of time. I'm electing not to re-prime with the used primers, as I want to be able to count on my loads 100%.
Thanks all, for the ideas and tips. That's why this board is a great one.
R.