ATT Arris NVG599 set up for IP Passthrough to Airport Time Capsule but connecting any devices to ethernet ports on the TC brings down internet service.

I don't have ATT service or an NVG 599 to test, so you might want to wait for another user to reply who might have the same kind of setup.

I assume that you previously reset the TC back to factory default settings.....before....you connected it to the ATT router.....so the TC would have picked up new settings from the NVG router.

Meanwhile, we can check the IP Passthrough on the NVG router to make sure that the TC is getting a valid Public IP address.

Please open up AirPort Utility on your Mac, and click on the image of the TC, and a smaller window will appear. Look for the IP Address in the small window and post back with that info. Please do not post the entire number, just the first 4-5 digits. For example, if the IP address reads 88.62.121.28, post back with something like 88.121.x.x.

Next, look for the LAN IP address of the TC and post back that info as well. If the TC was reset back to default settings before you set it up with the NVG router, the LAN IP address will read 10.0.1.1.

Next, click on Edit in the smaller window and more settings for the TC will appear. Click the Network tab and report on the current setting for Router Mode. It will likely be either DHCP and NAT or Off (Bridge Mode).

Next, click the Internet tab and report on the setting that you see for Connect Using. It will likely be DHCP.

Finally, a note to say that the top speed on the TC when it acts as a router will be in the 300-350 Mbps range on a wired Ethernet connection. Not sure what speeds that you are seeing at the present time, but speeds can be checked later after the TC is working correctly.

That AirPort Extreme that is "extending" the network.......

Open up AirPort Utility, click on the AirPort Extreme, and then click Edit in the smaller window

Click the Wireless tab

The setting that you want for Network Mode is "Create a wireless network" that uses the same wireless network name, same security and same password as the TC network.

You likely have the AirPort Extreme set up to "Extend a wireless network". That is not the correct setting.

That setting is used only if the AirPort Extreme is connecting using wireless. You are connecting using Ethernet, so you must use the "Create" setting to avoid the massive feedback loop that will occur if you are using the "extend" setting and then connect to the AirPort using a wired Ethernet connection.

Click the Network tab and make sure that the setting for Router Mode is set to "Off (Bridge Mode)

Click Update at the lower right of the window

Check the network.