Samsung Galaxy S2 wifi - Connected but no internet
My solution was to change the front room wifi to WPA2 to match the backroom's wifi (WPA2) which I never had a problem with.

Don't know why this works I cannot believe the SGS2 has less than optimum support for WPA. Maybe it was buggy software ... although I am fully updated and have been since I got the phone. (September 11)

I should mention this was not a problem caused by ICS or fixed by the firmware upgrade as I had the problem pretty much as soon as I got the phone (September 11).