Sorry, one more update. But this does narrow it down. I deactivated the "cbpaidsubsbot" Joomla plugin, and now we can log in again. This was after trying every CB Subs setting and plan setting that would freely allow anyone to log in. All the gates are down, as far as I can tell. I'm missing a setting or key functional understanding somewhere. This is a recent phenomenon; it all worked a week ago. We have 10 plans that are all set up the same general way, and the subscriptions that I've been using to log in are all active and have six more months before they expire. We also have allowed free subscriptions and lifetime subscriptions, none of which could log in.
At least I originally posted in the right forum category, I guess.
By the way, caching is completely disabled, not that it matters anymore, and I'm using Joomla 3.6.2, CB 2.0.15 and CBSubs 4.1.0.
If you'd be willing to log in, I have a test site that can be manipulated any way you like.
Thanks, guys.