Can a PS3 slim play ps2 games?

In short the answer is NO. Only the PS3 60 GB can play PS2 games. This is the only model of PS3 that has backwards compatibility with the PS2.

The answer is still no, but more PS3 models can play the PS2 and the details follows:

The 20 GB PS3 could play PS2 games and so could the 60 GB model that was released with it and those machines are called the first generation PS3 Models. The second Generation PS3 model were the 60 GB Model and the 80 GB model and while they could play the PS2 games it was based on Partially software-based PS2 emulation while the earlier models were Hardware-based PS2 emulation. These 1st and 2nd generation PS3 models all included 4 USB ports and also had Flash memory card readers. The 3rd generation of PS3 models had the 40 GB, the 80 GB, and the 160 GB Harddrives but only 2 USB ports and do not play PS2 games with backward compatibility. The fourth Generation of PS3s are the 120 GB and 250 GB PS3 slim model which do not play PS2 games with Backward Compatibility. All PS3 models play PlayStation 1 Games and do so through software emulation.