Reporting back:
I may have figured out the problem. I previously maintained two Cox Cable email accounts as well as a Fios one. I had two email accts for myself with Cox, since Cox would not let the "main" email account be named the way I wanted. But once I set up the "main" email account, I could set up a subsidiary account named the way I liked.
So I had 3 active email accounts for several months, but only one was used for sending. I suspect that Cox had some software that "phoned home" every now and then via email, leaving no trace of itself (Cox was my previous Internet provider, now Fios). I say this because I have never seen "sending 3 of 3" since I canceled the Cox account. I think I was seeing the reporting of the message I was sending, and one each for the two Cox accounts reporting in.
Anybody know if this "reporting in" could be so?
If I ever see the "3 of 3" message again, I'll report back.