Your setup is wrong. (if you are using the terms correctly)
You cannot extend the wireless of the ATT wireless gateway.
Plug the WAN port of the TC into any LAN port of the ATT wireless gateway.. do a factory reset and start over.
Here are the basic instructions for yosemite.. which is a problem but it helps with earlier OS as well.
Factory reset universal
Power off the TC.. ie pull the power cord or power off at the wall.. wait 10sec.. hold in the reset button.. be gentle.. power on again still holding in reset.. and keep holding it in for another 10sec. You may need some help as it is hard to both hold in reset and apply power. It will show success by rapidly blinking the front led. Release the reset.. and wait a couple of min for the TC to reset and come back with factory settings. If the front LED doesn’t blink rapidly you missed it and simply try again. The reset is fairly fragile in these.. press it so you feel it just click and no more.. I have seen people bend the lever or even break it. I use a toothpick as tool.
N.B. None of your files on the hard disk of the TC are deleted.. this simply clears out the router settings of the TC.
Setup the TC again.
Then redo the setup from the computer with Yosemite.
1. Use very short names.. NOT APPLE RECOMMENDED names. No spaces and pure alphanumerics.
eg TCgen5 for basestation and TCwifi wireless name.
If the issue is wireless use TC24ghz and TC5ghz with fixed channels as this also seems to help stop the nonsense. But this can be tried in the second round. ie plan on a first and second round of changes to fix this.. hopefully.. I will point out other steps that can be round2.
2. Use all passwords that also comply with 1. but can be a bit longer. ie 8-20 characters mixed case and numbers.. no non-alphanumerics.
3. If the TC is main router you can skip this point. This is only an issue when the TC is bridged.
Ensure the TC always takes the same IP address.. you will need to do this on the main router using dhcp reservation.. or a bit more complex setup using static IP in the TC. But this is important.. having IP drift all over the place when Yosemite cannot remember its own name for 5 min after a reboot makes for poor networking.
4. Check your share name on the computer is not changing.. make sure it also complies with the above.. short no spaces and pure alphanumeric.. but this change will mess up your TM backup.. so be prepared to do a new full backup. Sorry.. keep this one for second round if you want to avoid a new backup.
5. Mount the TC disk in the computer manually.
In Finder, Go, Connect to server from the top menu,
Type in SMB://192.168.0.254 (or whatever the TC ip is which you have now made static. As a router by default it is 10.0.1.1 and I encourage people to stick with that unless you know what you are doing).
You can use name.. SMB://TCgen5.local where you replace TCgen5 with your TC name.. local is the default domain of the TC and doesn't change.
However names are not so easy as IP address.. nor as reliable. At least not in Yosemite they aren't. The domain can also be an issue if you are not plugged or wireless directly to the TC.
6. Make sure IPv6 is set to link-local only in the computer. For example wireless open the network preferences, wireless and advanced / TCP/IP.. and fix the IPv6. to link-local only. Do the same for ethernet if you use it.
Now in this setup the TC will be in bridge mode.. and you will create a wireless network.. you cannot extend.. that will cause continual network crashes.
If you still find it unstable swap the connection now that it is setup from WAN port of the TC to a LAN port. The AC model airports have lousy WAN ports for some reason.. in bridge the LAN ports can be used instead.
also has a airport express that extends the network from the Time Capsule.
Also reset the express and configure it.. extends the wireless from the TC is fine.. as long as there is no wired connection.
The airport utility will pick this configuration by default.
If this continues to give you trouble.. then the TC could well be faulty.. it is worth taking it back to apple store if under warranty or applecare for testing / replacement.
