I have connected a wireless router to my computer and my Tivo, which are working fine. When I connected an ethernet cable from the wireless router to my PS2, I tested the connection through the PS2 and it was "successful", but when I try to log in, it tells me that a network cable is disconnected, even though I have over-rechecked both ends of the cable, and the wireless port is lit up where I connected to it. Do I have to turn everything off and connect while the router and computer are off? My friend said he just plugged and played, but I have had no such luck.
If there is a way to troubleshoot connection on the PS2 do that and if it says network cable disconected then all I can think of to do is turn off everything, tivo, computer, PS2, modem, and router. Also unplug all ehternet cables. Start reconecting all ethernet cables then power on in this order. Modem, router, computer, and then Tivo or PS2. Doesn't matter on the last two. If that doesn't work, you might check that it is a high speed ethernet cable.
At my house I also have a router, and when I have a friend over we just connect our 360's via wireless adapter. We have also done wired. In that case we have just plugged and played. Of course we have subscriptions to Xbox Live and have a profile and all that. So make sure you have a PS2 online profile.