Just the same problem i have with my SGS2, have to try to re-enable my wifi few time to get it have access to internet tho its show connected.
at first i tot its my linksys router DHCP problem so i tried static (I used static for my laptop too).
Screen shot of my SGS2 screen.
and i have do a manual verify that sgs2 show connected but in router it is not as it doesnt show in the ARP Table. after a few turn on-off of SGS2 wifi and the IP address dedicated to SGS2 show in ARP Table the I can internet on SGS2.