I live in what use to be a hotel and is now efficiency and the plumbing drains runs from 2 units to one drain line. If 1 drain is backing up wouldn't both drains back up?

Depends where the clog is. A clog in one unit and not the other indicates a clog before the two lines meet both lines being equal to the main.

I agree with talaniman. It will only back up in both if the clog is beyond the point both merge into one.

Yep, correct. If the clog is upstream of the Y connection only one would backup.