Whenever I hear of people having problems with this type of stuff I automatically say it's a bad router.
Why? Because the game is the same for everyone, it's just a disc your PS3 runs. Now it may have something to do with port forwarding but the fact that it disconnects you back to he main screen and also screws up the internet for everyone else, it leads me to believe your router is toast.
There's no reason this should occur barring one of your family members leaving uTorrent open with no up/down restrictions
I would try a new router. I always had problems with Linksys in the past so I decided one day despite problems I had with D-Link in the past that I would give them another shot.
Boy am I ever happy I did. Their routers are sooooo much better now. Will never buy a linksys product again and if I do it'll be Cisco. Cisco just doesn't seem to put the same kind of effort into their Lynksys products as they do with Cisco products.
Seeing as how D-Link has worked brilliantly for me for years now without any ********, I'm sticking with them.