Trouble connecting iMac directly to modem

You have removed the Airport Extreme??

If you plug the computer directly into the modem, does the ethernet light come on, on the router. That is essential for ethernet to work.

Are you using an ethernet cable you purchased? If it is super long they can give trouble.. a short patch cord should be fine.

If you are unsure plug the computer by ethernet into the Extreme using a short patch cord… does the little light on the AE turn on.. if yes.. you can plug the cord from the cisco to another ethernet port on the AE and make sure it is also working.

Network Preferences tells me "Thunderbolt Ethernet has a self assigned IP address and will not be able to connect to the Internet." Sorry, not sure what this means.

Either ethernet hasn't connected or DHCP is not working. If the modem was bridged it will not work, unless you power off the modem for about 20min. Then power it up again followed by the computer.

Can you connect the Ethernet cable from your Mac to one of the three LAN <--> ports on the AirPort Extreme?

What happens when you do this?

I have an AirPort Extreme providing a wifi network in my home. It is connected to a Concast Cisco modem model DPC3941T.

The Cisco DPC3941T is not a simple modem, but actually a combination modem & wireless router (also known as a gateway) device. If you have the Extreme connected to it directly do you have them both configured for a roaming style network? This type of network allows for each wireless access point to broadcast its own Wi-Fi network, but since each will use the same network name (aka SSID) and use the same wireless security type & password, wireless network clients will see them as a "single" network. Is this the way you have them set up? If not, then this may be why you are having with the Wi-Fi dropping out.

Are all devices dropping out, or is it just the Mac? If other devices are working OK with wireless, then it is possible that connecting the Mac to the AirPort using a wired connection will solve your issues with the Mac. Might be worth a few minutes to try this.

You can still find out whether you have both a WiFi and wired problem on the AirPort Extreme. By that I mean if you connect the Mac to one of the Ethernet ports on the AirPort Extreme......and it works.....then you have at least narrowed the issue down to the wireless on the AirPort router....and we can start to troubleshoot that issue.

As it stends now, we don't know what the real issue is, so we're trying to suggest ways to find out what the real problem is......rather than guess at things.