Ok.. that is a good start.
Have you recently loaded any software on the computer?
eg.. any of those terrible adware apps.. cleanup, startup, speedup.. etc .. almost anything with UP in it.
Or security/anti-virus software??
You are using the TC in bridge.. and your modem is also a router.. just so you know.
The very fast failure of the TC from the network is often a result of poor compatibility with the main modem router.
So a couple of things to do.
1. Where you now have the TC plugged into the modem router via its WAN port.. I want you to change that over to LAN port. So to make it easy in pictures..
Here is mine.. the cord to the main modem is blue one.. and that is the WAN port at the bottom.
The yellow cord goes to the computer.. that is LAN port.. unplug the cord from the WAN port.. then plug it into any free LAN port.

Then shutdown the whole network..everything off.. then restart in correct order.
Modem.. wait 2min
TC.. wait 2min
computers and other devices.
2. If this does not help then I need you to also redo the setup of the TC.
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 you want to use for TM using Airport Utility.
1. Use very short names.. NOT APPLE RECOMMENDED names. No spaces and pure alphanumerics.
eg TCgen5 for basestation and TCwifi wireless name.
If the wireless is still a problem set wireless names like TC24ghz and set the 5ghz to different name.. eg.TC5ghz
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.
Ignore this point if you don't understand it .. post back if you have drop out issues.
4. Make sure IPv6 is set correctly in the computer. ie link-local only for ethernet and/or wireless depending on which you use. Auto is not good enough.
If you cannot follow the above.. get a friend who is good at Mac stuff to help you.
In the end you will need help if you have issues understanding it.. because the problem is not magically going to go away.
3. If you have any applecare on the computer then ring apple support and get them to talk you through it.