Unable to connect Time Capsule to existing network

Connected TC WAN port to ATT modem

I am guessing that the ATT modem is a router.. please confirm.

Selected TC in Airport Utility. Page says This Airport Time Capsule will create a network. Selected Other Options. Selected add to an existing network

Let the TC do its thing.. create a network is actually correct.

You are not adding or joining it to existing network.. that generally means by wireless.. and Airports don't work that way.

So finish the wizard .. ie you put in your airport name, wireless name (identical to the existing SSID).. wireless password same as existing network. Main password.. make it different to the wireless if you like.

Once finished and the TC will reboot with new values.. you can then check it is working correctly.. if it fails again.. you need a computer with ethernet port. I suspect your Mac doesn't have one but you can get Thunderbolt or USB to ethernet adapters.. it is difficult using wireless to configure wireless.

The TC is going to need to be permanently connected to the ATT modem/router using a wired Ethernet cable connection.

Go ahead and set up the TC to "create a wireless network". If you don't want to have 2 wireless networks, you can later turn off the WiFi function of the TC and use the ATT wireless. Because the TC will remain connected to the ATT modem/router using an Ethernet cable connection, the ATT WiFi signal will be transferred to the TC.

So, while it is not exactly correct to say that the TC has joined the WiFi of the ATT modem/router, in effect it has become a part of the ATT network.