Sure, its possible... I mean we're not talking a Steinway D, or anything. But you should be able to find a piano with a pleasing tone, even action, and which holds a tuning for $1000. I recently looked at an Aston-Weight from the 1950's, for a friend and her daughter. The seller was asking $1000 and I thought it was a nice piano for a beginner/intermediate player. It actually made me feel kind of silly for spending $16,000 on my grand (OK, I was starting to feel silly before that).
Whatever you do, don't get a spinet!
There is a certain amount of risk buying an old piano, or even getting one for free. If it turns out that, say, the pin block is shot and it won't hold a tuning, you're stuck with a large, heavy piece of junk.
Oops, now I see this post is kind of old. Well, maybe it'll be useful anyway.