OK, booted from an old, known good system and it still failed. I did continuous pings to the DNS and it failed after 107:

So, it ain't software....

So it seems (from the traceroute) that your machine can send packets through the switch and get to the basestation but then they get lost - they don't seem to be routed out of the basestation. Very strange.

I'm about out of ideas. Only one thing occurs to me - is it possible that this trouble is caused by some periodic activity on one of the oteh rmachines connected to the basestation? Try disconneecting all but the one machine and see.

My cable modem is back running. I can run the Ethernet cable directly to the once troublesome machine and it now stays connected to the Internet with no trouble. So, I have to assume now that there must be something wrong with either the Ethernet switch or the ABS.

However, if I route the Ethernet cable from the Cable Modem to the Airport so I can use the ABS as a router and bridge, I am unable to get ANY machine to connect to the Internet. (How odd that it worked for two out of three machines when the ABS was doing the dialing.)

Here's how the ABS is set up:

INTERNET:

Connect using Ethernet.

Configure using DHCP.

The ISP's DNS servers listed.

NETWORK:

"Distribute IP addresses" is checked.

"Share a single IP address (using DHCP and NAT)" is selected.

"Enable DHCP server on Ethernet" is checked.

"Enable AirPort to Ethernet bridging" is checked.

Here's how each individual machine is set up in the Network preferences pane:

Configure: Using DHCP

Is there something wrong with the way I'm configuring either the ABS or the individual machines? I'm sure this is the way I was doing it before the Cable Modem went down....

Your config sounds reasonable.
Are you sure you have the cable modem coming into the WAN port (and the switch on the LAN port)?

is it possible that your cable company registeres the MAC address of the machine connected to it and is refusing to service a different MAC address (your basestation)? The usual fix for this is to turn off the cable modem for a while - maybe a few hours - to let it forget about the old MAC address. Or call your cable company and ask them to check what the problem is.

I should add here that when I am trying to use the ABS as a router/bridge I cannot ping the Cable Modem either at the address assigned by the ISP or at the address it uses for web based configuration. I can ping all the other machines on the network.

I should also note that I tried this with only a single machine on the LAN with no success.

Yes, the Cable Modem is connected to the upstream port on the siwtch. The ABS and the wired machines are all on the other ports.

This is how it's always been.

I'll try shutting off the cable modem while I go out and run and I'll get back to this tomorrow morning.

Thanks to all for your help and suggestions.

Sorry. This didn't work.

Maybe I'l abandon the ABS as a router and use the OS/X one....

But this didn't work. Here's what I did:

I reconfigured the ABS to serve only as an Ethernet/Wireless Bridge.

I left the Cable Modem connected to the WAN (upstream) port on the switch.

I enabled Internet Sharing on the troublesome iMac.

I expected the iMac to get an IP address from the DHCP server in the Cable Modem, just the way the ABS does. But, the iMac can't seem to get an IP address from the Cable Modem.

What am I missing here? If I connect the iMac directly to the Cable Modem it gets an IP address just fine. Why can't it get one when it's connected to the Cable Modem through a switch?

In the last example your mac still gets its iP from the ABS, this is where the problem lies.

Shut down all hardware except the ABS, then disconnect everything.

Hard reset the ABS (check manual for location, ABS will forget all settings, so note down first if necessary)

Power down ABS

Connect the CM to the WANonABS, the LANonABS to your Switch, Your Switch to TM (troublesomeMac).

Power up the CM (let finish checks)
Power the ABS (ditto)
Power up TM

Use TM to reconfigure ABS WAN and LAN, Set ABS as DCHP server, leave off all Encryption, Access controls and any other options, You may need to clone your MAC address from TM to ABS.

If procedure fails, Hard reset ABS again & reconfigure.

If procedure fails second time remove (better to replace with another switch if possible) switch & try again.

IF (CM____ABS____TM is False) AND (CM____TM is true) then ABS = FUBAR

dD

Whoa! All this time I'd been assuming that your switch was connected to the LAN port of the ABS and that the only thing connected to your cable modem was the ABS. Something like this:
Note: the upwards side of the ABS and the switch are the "WAN" connections. The downwards side are the "LAN" connections.
That's the way I would do it. Your ABS is the gateway to the Internet. All of your 3 macs get their IP address from the ABS's DHCP server.
But you seem to have it like this:
It is not clear which port (WAN or LAN) you are using to connect the ABS to the switch. (It should be the WAN side.)
This means that mac2 & mac3 are talking directly to the cable modem without any NAT. I don't see how this could work. Anyway, there's not much point in thinking about it since I don't see why you would want this configuration when you could do it as in the first diagram which is far more secure.

All of this is assuming that it really is a switch that you are using and not a router-switch combo.

Sorry folks, this is an old graphite ABS with only a single Ethernet port. This is the way it has always been set up and has always worked just fine:


The single port on the Cable Modem is connected to the upstream (WAN) port on the switch. The ABS and the wired Mac's are connected to downstream (WAN) ports on the switch.

The ABS has always been able to do DHCP and NAT configured like this and I think I got the configuraton from an Apple document called "Configuring Airport Networks".

The ABS gets it's IP address from the Cable Modem. When this was working I was using fixed IP addresses for the Mac's and the ABS was doing NAT. Perhaps I ought to go back to fixed IP addresses for the Macs to see if that works.

have you tried this?

dD

Hmm - I wouldn't have thought this would work but I guess if mac2 & mac3 are configured to gateway through the ABS then it would be okay.

I can see a possible problem with DHCP since mac2 & mac3 are going to see the DHCP server from the cable company. I'm not sure what determines which DHCP server gets used when there are two of them on a LAN. (Your machines mac1 & mac2 are on the cable company's LAN since there is no router between them and the cable modem.)
And there might also be problems if you give your machines static IP addresses that collide with those assigned by some of your neighbours - whose machines are also on the cable company's LAN.

Summary - I still think there are big problems with the config you have outlined above due to the fact that mac1 & mac2 are on the cable company's LAN.
It would be far simpler if you bought a router box to replace your switch.

If memory serves me correct, you can use the older airport base stations as a router without having it do DHCP. The Extreme base stations don't do NAT without DHCP. Anyway, disable DCHP on the base station and use fixed IP's for all your machines. This will avoid any DHCP conflicts with the cable modem. I had one of the older base stations and had the same setup and this worked fine. A better option would be to get a router to put between the cable modem and the switch.

It Was The Switch
 

I replaced the suspect switch, set up the LAN exactly as described before:

and all is well.

The only difference is that now, at the behest of several posters in this thread, I am using DHCP rather than fixed IP addresses for all the machines on the LAN.

It seems to work like this: The cable modem serves up an IP address (202.128.1.n) to the ABS. The ABS serves IP addresses (10.0.1.n) to all the machines on the LAN. The ABS does NAT and bridges the wired and wireless networks.

Thanks to all of your for your help and patience.