I can confirm what everyone is saying here but I got it to work correctly. I have a 4th Gen Time Capsule and a 5th Gen Airport Extreme. I was having the same issues after updating the firmware to 7.6.3, ie, drop outs with both wired and wireless connections, slow downs, etc. The final straw was when It quit sending out DHCP renewals to my devices and they all fell off the network. (Not sure how relevant it is, but like other people mentioned, I too have both dhcp reservations and port mappings in the Time Capsule configuration.)
Just for record, I used Airport Utility 5.6.1 to do the following:
The first thing I did was to load up airport utility and export the configuration files for both the Time Capsule and the Airport Extreme to my desktop. The second thing I did was to unplug the power to the Airport Extreme to simplify troubleshooting. The third thing I did was to perform a hard reset on the Time Capsule by using a paper clip to hold the reset button in on the back of the Time Capsule for 5 seconds until the light turned amber color and blinked fast.
After that I let the Time Capsule boot up completely again and went back a workstation and started the airport utility. After joining the generic apple wireless network (apple2sdfw or something) I configured the Time Capsule with the default settings it recommended for Time Capsule name and Wireless network. After I updated the Time Capusle with the new settings and it rebooted everything worked fine and I had no issues. (other than I lost all my previous settings for wireless network, port mappings, dhcp reservations, etc) I ran a few speed tests and everything was great.
So then I imported my original settings for the Time Capsule and updated it and when it rebooted it acted just like before and wouldnt give out a IP address via DHCP to either wired or wireless clients, although if I put a manual address in the tcp/ip settings on the workstation it would work fine.
I went back to the Airport utility and under Internet/NAT I unchecked Enable default host and also unchecked Disable NAT Port and then clicked on Configure Port Mappings and unchecked all the items I had listed in there. Lastly, under the DHCP tab I removed all the entries I had under DHCP Reservations and then updated the Time Capsule.
When it rebooted it still wouldnt hand out a DHCP request so then I went into Internet/DHCP and changed the DHCP range from 192.168. to 10.0. and under Airport/Wireless changed the wireless name to something random I had never used in the past and then updated the Time Capsule. When it rebooted everything worked perfect and it handed out a DHCP address to my clients. I ran more speed tests and everything was working fine.
Next I just went back into the Airport Utility and changed the DHCP back to 192.168 and then updated the Time Capsule again and everything still worked great so I changed the wireless name and updated the Time Capsule again and it still worked fine.
Again I went back into Airport Utility one last time and re-enabled the Default host and set the IP address for it and then re-enabled the NAT Port Mapping Protocol and clicked on Configure Port Mappings and put check marks back on all my port mappings. Under Internet/DHCP I manually entered my DHCP reservations for the clients I had reservations for and then updated the Time Capsule for the last time and it has worked fine ever since.
Hopefully this helps someone out... You may be able to even cheat a little and import just some of your settings again back into the Time Capsule after doing a hard reset but Im not sure about that. Personally, I think it was the wireless network name that was screwing everything up, it seems like once I changed my wireless network name it seemed to fix the problem.