I live in a downstairs unit in a condominium complex. My kitchen sink started backing up one evening, but after working on it, I was able to get the water to go back down. The next morning, after running the water, both sides began to stop up. Once again, I got it to go down. About 1 hour later I walked into my kitchen and both sinks were full to the brim. I dumped the water out and plunged both sinks and put in Drano. The water eventually went to where I could still see some inside the drains. I left for about 7 hours and later that night my sinks overflowed. A plumber was called and he unstopped the leak. The plumber told me that it was a line and “it was not your fault.”

If I was gone and the water backed up into my sink because someone else was running their water, what could have caused this?

This comes under the heading of a "Common Area" problem. Since the house sewer mains are used by all the condo owners the condo association should pick up the tab to have the line snaked and cleared.

"If I was gone and the water backed up into my sink because someone else was running their water, what could have caused this?"

Because your unit is down stream, and under, from all the units above it when the main gets blocked EVERYTHING that is drained or flushed from above will end up on your units floor. And what's worse, if you're gone, nobody will kinow anything's wrong and continue discharging sewage until it runs out your door. I've had this happen in a three story condo in my area. It cost $27,000 to rebuild the unit. Everything was trashed that could soak up moisture. We wouldn' have known anything was wrong since the owners had gone back North for thr summer season. A maid noticed the water leaking out the front door.
I tell you this so you'll be aware of can happen on first floor units. Good luck, Tom