Having just received my WIFI Kindle3 I was slightly terrified to discovered that it seemed unable to connect to my Apple AirPort Extreme base station, the Kindle continually just said it could not connect.
It turns out that “connecting to WIFI” on the Kindle actually means “connecting to the Kindle Store using WIFI”. We had just moved house and were waiting for our internet to be connected, and I assumed that it would at least connect to the network but then report an error if you tried to use online services.
So if your internet connection is not available, it will connect to the WIFI network, fail to find the Amazon site and then disconnect with a non-specific network error message.
Googling around I discovered lots of very confused forum threads suggesting a range of fixes from installing the preview (ie beta) Kindle software update, to needing to use the 64 character pre-share key from the AirPort to authenticate if you were using a WPA or WPA2 network. Once person even claimed that Amazon support said that the AirPort Extreme was not supported. I don’t know if any of these people had different issues but none of the suggested fixes helped me at all, it was purely related to internet connectivity.