These days, there are only certain models that can read the PS2 games from the disc drive. They are the bigger ("fatty") PS3's that were made around launch.
Even then, not *all* fatty's play PS2 discs. Personally I still have my MGS4 bundle PS3 which does not play PS2 discs.
The slim and super slim (super slim referring to this PS3 model) do not play PS2 discs.
HOWEVER, if you turn on your PS3, log in to your PSN (I'll never get used to calling it SEN) and launch the PS Store, there are some select downloadable PS2 titles (such as God Hand, Max Payne, etc) usually on sale for about $10. But, this isn't the entire selection; studios are finding it a little more profitable to re-release old games as "HD Editions", like Metal Gear Solid 2 + 3, the Devil May Cry games, etc.
I'm sorry I can't answer the question directly with a Yes or No. Yes, some older systems can play the discs (later ones, no), yes that there is a selection available as downloadable games (but no on every single PS2 game being available).
I hope this helped out!
Some references: http://en.wikipedia.org/wiki/PlayStation_3#Model_comparison http://en.wikipedia.org/wiki/List_of_PlayStation_3_backward_compatible_PlayStation_2_and_PlayStation_games