When you say you "flipped" the breaker, I presume you mean you turned it off. What kind of tester was your handy guy using? Non-contact voltage testers (sometimes called a "volt tic") and digital multimeters can give erroneous readings in some situations. Also when testing a circuit with a multimeter or solenoid type tester it's important to test from hot to neutral and hot to a known good ground. No power from hot to neutral but power from hot to ground would indicate a bad neutral.

You might be best served by hiring a licensed electrician with experience in older residential wiring. With the age of the house and assorted wiring methods in use, this might be a difficult problem to troubleshoot that will require someone with experience in older wiring methods.

While you are measuring the voltage add a load.

Old house wiring can be tricky. I remember reading years ago about a case where an old house's cellar light worked in the winter but not in the summer (the opposite of your problem, I know, but this is really just for the information and entertainment value).

It turned out that the electrician who'd installed the original knob and tube had missed soldering a splice. When the copper wire contracted in cold weather, it pulled the splice together enough to make the light work. In warm weather the wire expanded and sagged a bit; the splice loosened up and the light wouldn't come on.

Not saying this is your problem, just that troubleshooting old houses can be an interesting challenge. Sometimes an old-timer can be helpful for these cases.

That sort of thing IS possible.

I'm thinking there's some sort of automatic sensor or otherwise, probably related to the heating/cooling systems, that has been miswired.