I just upgraded my second computer to Windows 7 Home Premium, a Dell Inspiron 1525 With a 2Ghz Core Duo and a Dell Wireless 1395 WLAN Mini-Card. Initially, it was having wireless problems.
It would indicate that I was on the network, and indeed, I could access local resources. It would also say that I was on the Internet...but I wasn't. It also identified my network as "(SSID) 2" as opposed to "(SSID)" as my other machines properly do.
I uninstalled the network adapter to force the OS to discover it again. That cleared up the problem.
I hope this information might help someone else avoid the frustration.