Yes they can and they can also play on the new slim released in May 2010. They really are only PS3 games and not games for the slim or fat PS3
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No the new PS3 slim can not play PS2 games. To make the PS3 profitable and have it quit selling for a loss Sony stopped making PS3s that could play PS2 games long before the release of the PS3 slim. The only reason you could ever play PS2 games on the first PS3s was the game catalog for the PS3 was next to nothing. Once the catalog of games increased they dropped the ability to play PS2 games.
At the moment, no. Sony has unfortunately cut PS2 backwards compatibility, and the "Other OS" capability from the PS3 Slim.
In short, the 250GB PS3 Slim console WILL play PS1 games but WILL NOT play PS2 games. No PS3 model with 2 USB ports will play PS2 games and can not be made to play them through hardware or software updates or any other changes The 120 GB and 250 GB are older models of PS3 Slim and the new Slim Model PS3s still do not Play PS2 games, but have 160 GB and 320 GB harddrives and sell for $250 and $300 in the USA Every PS3 will play PS1 games. Only the 20GB, 60GB, and the early 1st 80GB PS3s will play PS2 games. They all had 4 USB ports
the playstation 3 250 GB can not play playstation 2 games
No the only PS3 consoles that play PS2 games are the early PS3 models with 4 USB ports as can be seen on the related link to the PlayStation Knowledge Center.