The only PS3s that can play PS2 games are 20GB/60GB PLAYSTATION®3 and 80GB PLAYSTATION®3 Model CECHE01. *If your model number is not CECHE01, then it will not play PS2 games.
Here's a better list. It says any CECHE model will work for PS2 backwards compatability. *These models are not being made anymore, so if you just got it then you probably do not have backwards compatability with PS2 games.