The Short Version: I can make it work by selecting “Connect to Xbox Live”, letting it fail the test, and then choosing the option to test the connection to Xbox Live. After it completes successfully I can then connect to Xbox Live. I never found a solution to this problem.
Rather than deal with this problem indefinitely I removed the Microsoft adapter and now I’m using an ethernet to wifi bridge device, which works perfectly.
Several months ago I moved the Xbox 360 into a different room. Rather than run another network device for the only console that doesn’t have built-in wireless (I have an older model) I decided to purchase an Xbox 360 Wireless Network Adapter.
It works, but not perfectly. I haven’t identified the cause yet. It could be a compatibility issue with my Apple Airport Extreme Base Station though it may be something very obscure. It doesn’t receive a very strong WiFi signal but I wouldn’t rate as being weak.
Whenever I start the Xbox 360 up it no longer automatically logs into Xbox Live (despite being configured to do so). In addition, it won’t connect to Xbox Live just by clicking the appropriate tile.
The only way I can make it connect using the WiFi adapter is to select the option to test the connection to Xbox Live, after it fails to connect.
After the test successfully completes I can then back up to the main menu and sign into Xbox Live without problems.
Updated 06/25/2012: I still haven’t found a permanent fix for this issue. It’s as though the USB wifi adapter simply doesn’t wake up when I turn on the Xbox 360. I have to run a connection test every time to get it working. The issue isn’t caused by the Airport Extreme Base Station. It took a hit to some ethernet ports recently and is no longer in line. I’m still experiencing the same problem with my ASUS wireless router.
Updated 12/10/2012: I sold it to a friend recently and he reports that he doesn’t have this same issue. I suspect that the wifi signal where my Xbox system is located is rather weak, which creates problems for the Xbox adapter. I’m currently using an IOGear device with the Xbox and it seems to be able to connect to Xbox Live automatically at start-up.