I recently left a sink on for about 10 minutes in my bathroom. The sink was stoppered so water overflowed the sink. I cleaned up the water immediately; meanwhile the downstairs tenant had water leaking in and contacted the landlady--the problem downstairs was also dealt with immediately. My question is should that little water cause a lot of damage? I was surprised the water leaked through the floor so quickly.

Yes, a little water can cause a lot of damage. If your faucet has a 1 gallon per minute flow, you probably put 8 gallons down stairs.

Harold's correct, a little water goes a long way. Water will always seek the path of least residence so it got from your floor to the ceiling below very quickly, There's no option here. Just be happy the damage wasn't too great.
I was called on a call like yours.
This was a vacation condo and they left for up North, IT ran for some time and the damage ran to $20,000 after we replaced everything from the bare walls, ( and I mean stripped of drywall and insulation) to every thing that wasn't made of metal.
I'm sorry it happened to you but it could have been much worse.
Good luck. Tom

Got one worse than that. Toilet tank fill line started leaking at a lake house. Guy was gone for a month, 350,000 gallons ran onto his 3rd floor. House was totalled, $379,000, insurance paid everything except his $3000 waqter bill.

Bob,
Can I use that one when I am trying to convince the wife that we should turn the water off when we go away?

Sure, but can you believe this same guy still doesn't turn his water off when he is gone.

Thanks for the help! I was thinking a bathroom floor should be able to withstand some water, given that there are 2 faucets in it. Sigh.

Turning your water off when you leave on vacation is the cheapest insurance you'll ever get. Bottom line? It's simply dumb not to!
Cheers, Tom,