old iMac having trouble connecting-HELP!

I have 2 old white 2006 iMacs, a 24" and a 20"

In a recurrence from a week or so ago, the 24" woke up with no connection. After a reboot, it was OK. But today, the same thing. Network diagnostics said to reboot the modem, which I did although it seemed fine. Rebooted the iMac as well. Still nothing. Went into the settings, said no IP address, but on further poking doesn't see any WiFi networks at all. Says ethernet cable is disconnected although I plugged and replugged it.

Hmm. Connected the 20"-WiFi and Ethernet are fine (I removed the cable from the 24" and plugged into the 20")

Uh oh.

Seems to me like something in the hardware is getting fritzy??? Any ideas? Otherwise, the machine seems to run fine. I do NOT want to get a new, reflective-glass-screen iMac, since this thing has a window behind it. And I'm not really too keen on getting a used one from eBay and tricking it out like this one...

So both Airport (WiFi) and Wired Ethernet Networking does nothing on the 24"?? What version of OSX is on the 24"? Seems strange that both would die as they are separate chips.

OK, this just got REALLY weird. My other 20" iMac now says I'm not connected to the internet, Network Settings failed.

I am typing this on the 24", which had the same problem even after reboot and modem reboot. This machine was asleep, I rebooted the modem per the 20" network assistant instructions. 20" still not connecting. Woke up this 24" for the heck of it...choose the network...connected.

The 24" is running 10.7.5, the 20" currently 10.6.8

I just don't know enough about network to figure what the heck is going on...and neither will U-verse Support...

Network Preferences says "No IP address" which is what the 24" said the other day.

And no, I am not getting confused as to which one had the problem previously!