I am having similar problems. I recently reconfigured my airport extreme base station from using a manual IP (from a different network) and brought it back home to set up using DHCP. I can get macbook to recognize the airport base station, and the signal strength is full. I will then open safari and the connection is great for about 30-60 seconds and then... nothing. It will cease to open any new pages after that short duration.
I have reset my modem and base station about 10 times. No change. When I link my macbook directly to the modem via ethernet cable, I have no issues. When I switch my macbook's airport to my neighbor's wireless network, I have no problems whatsoever.
I have tried doing a reset with the back switch and tried starting over by setting up a new wireless network in my home. I have a steady green LED on the front of the base station. If I turn my airport card on my macbook off, then back on, I get the same results - about a half minute of blazing connection... then nothing.
Can anyone offer any suggestions?