Thanks Judy,
I *might* do what you said (although it sounds rather onerous) however ...
As jb pointed out, Firewalls look like a culprit.
When my N360 Firewall was temporarily disabled I could ping, but not get full two-way access with other PCs on the network.
I *completely* de-installed Norton/Symantec (and I mean thouroughly got rid of each Norton/Symantec folder, service, registry entry ... following an "expert" guide") and ...
I could ping myself (unsurprisingly) but still other PCs couldn't see mine.
Then I installed the Free version of Zone Alarm, set up the IP addresses of other PCs on my network as being in the "Trusted Zone" and Hey Presto ! ... other PCs can now see mine.
I confess that I don't understand why the problem persisted when I had no Firewall whatsoever ... so I looked in Control Panel and found that the Windows Firewall, while ostensibly "off" had an advanced tab with a reference to LAN2 which seems to be enabled. Should I uncheck this now ? Or is that giving me some extra protection (without causing problems).
To any concerned reader - I have also installed Avast for anti-virus protection.
Thanks to you both
Graham
P.S. I am continuing with my "Admin Tools" issue in the Win XP forum.