It could be a partial blockage further down the drain line, possibly where the lines join each other below. I would have it snaked. You're only seeing it once in a while because the backed-up water is draining very slowly in the line.

I once worked on a basement apt where the shower drain would occasionally back up when the adjacent kitchen sink would get blocked. In that case, manually snaking it wouldn't solve the problem for very long (probably due to greasy materials in the line) so my last resort was to use a very strong chemical drain opener (take every precaution!!), followed by flushing the line with plenty of water.

If there is a main line blockage below the 1st floor tub the water from all fixtures above it will back up into the lowest fixture on the drain line. In this case it is the 1st floor tub.

A main line blockage would have made things easy to understand. I suspect the line may be partially blocked, but don't understand how this could have happened.

The tub was simply overflowing with water. We
drained it with a bucket and snaked the drain through the overflow tube about 12 feet. The tub remained clogged. It wasn't until we removed the strainer that we discovered a clump blocking the drain. Once that was removed, all was well.

We just don't understand how the tub filled up, and what the impact of the tubs on the second and third floor had on the overflow of the first floor tub.

It now sounds like you have/had two blockages. A main line blockage that possibly forced a second blockage into the tub drain. The tub could have filled through the overflow if the drain was blocked.

Yup, more than one blockage. The one you haven't cleared yet is the one that caused the tub to fill. It's probably in the main near where the branches from above link in. It's below the basement floor and 12 ft probably wasn't enough to reach it.

I agree, that sounds like a good theory...

Will take advantage of your expertise one more time. It's a cast 4" drainpipe that travels under the basement floor with no clean out openings until the point where the pipe leaves the house. That plug is located maybe 50' from the tub. There's currently minimal access into the pipe shaft.

The only readily available access to clear the drain is from the overflow pipe in the tub. Can I effectively clear a blockage in that size pipe with a 1/4" snake (powered by a drill) through the overflow pipe in the tub? Hesitate to use a liquid drain cleaner. What would you recommend?

Thanks again for the help.

Since you found that giant hair blob, I would be VERY inclined to use a chemical opener for the other clog, too.
Maybe just a dose of regular drain opener might do the trick -there's always next time if that fails.

As a last resort, after all else failed, I got the strongest stuff possible and follow the directions carefully. (I used sulfuric acid on very rare occasion as a last resort, which requires a respirator, proper ventilation, eye and face protection, long sleeves, etc..... Definitely really dangerous stuff, not for the average homeowner, but it worked after all else failed.

Possible alternative: If the 2nd floor is just a shower (not a tub), I might feed a long 3/8" auger through it. I have a 25' and a 50' manual for doing stuff like that.

Finally, there is the chance that the sinks in the bathrooms are still connected to the shower drain stacks. you might pull a trap from under the sink and go in from there.