OK. The issue is now resolved. I went through all the CB Subs, CB, and other components and disabled things one by one. I was about to give up, since I've got so many things running on the site, and it's a production site with 5,000 members.... Nothing was seeming to work, but I did start to notice a transient message and get a screen capture of it.
It read:
Unable to load listSubscribe()! Code=232 Msg=There is no record of the unique_id "" in your account
Googling on this pointed me to the JooMailer (MailChimp) plug-in.
The issue is mentioned at:
www.joomlamailer.com/topic.html?id=194
It appears that for any member who is not on a MailChimp list, the plug-in doesn't find them, throws this error, and then drops the save. (It would be better, of course, if they went forward with the save before checking the lists, and then adding them, or not adding them, so a failure would not stop all CB user modifications!)
Thanks for your help! Really love CB and CB Subs!