Exact same thing has started happening to me, tried everything. I know it is definitely the Airport Extreme that is causing the problem as I get full speed with no drop outs when connecting my Macbook to the modem directly over Ethernet.
The Airport Extreme I have is about 3 years old now so I have given up trying to fix it and ordered the latest ASUS dual band router that seems to get some good reviews.
The Airport Extreme I have is about 3 years old now so I have given up trying to fix it and ordered the latest ASUS dual band router that seems to get some good reviews.