Well, I got the problem fixed. It was weird indeed. Thought I'd post the results for others who might experience this problem. There seemed to have been 2 simultaneous problems occurring.
Tried a repair to 2K on the 2 machines that couldn't see each other. No go.
Tried ipconfig release/renew on both. No go.
Tried resetting the router. Nope.
Shut down both systems, unplugged modem, and the router for 5 minutes. Powered up and success!!
Or so I thought.
While both machines could now "see" each other, machine #1 still could not ping anything (including itself) and I discovered that Netscape could not connect to the net for anything - web pages or mail. (Use Netscape for mail) I.E. still worked.
So I "googled" error code 65 and found a MS doc saying that it was being caused by Zone Alarm.
This is the strange part. I do have Zone Alarm installed on my system but I had disabled it in startup in msconfig.
Thought to myself "If it's not running, how could it be the problem." I remembered that I had never given Netscape to access the internet without permission from ZA and decided to try opening ZA. Everything went back to normal including the ping tests.
Evidently, ZA must still run or have some effect even though it is disabled from starting.
Bill