I think this saga is at the end. Over the weekend, the backup behavior of both the XP and the W2k PC turned to total failure. Next, access to the unit thru the LaCie web (which is how you administer it) went from working fine to requiring a fresh login after any click (and always returning to the initial page).
At this point, I think the unit has lost it, and I am going to ask for a repair.
So, not a networking issue at all, only a hw/sw problem.
Thanks for the help and the forced thinking into the operation - removing some of the magic.
Tom