I've had many problems like this with the Wii trying to connect to wireless networks. I've yet to find any solution The rest would be 'successful', but the Wii would be unable to connect to anything.
It pisses me off really. My PS3, DS, and PSP can connect just fine, but the Wii can't?
EDIT: I've heard that changing your channels network from the default six to 1 or 11 instead helps, but I have never been able to try due to complications.