Maybe this is about to be solved.
I found APC's answer at http://www.apc.com/site/support/index.cfm/faq/
today. What I haven't tested for in their list is "[o]verloaded neutral wire (>5vdc measured between Neutral and Ground)." Checking their forum (it looks like if I post there I risk warranty coverage) in http://forums.apc.com/message/27798#27798
it appears I can test by measuring AC voltage using my multitester's AC scale without testing for DC in an AC circuit (as APC seems to suggest) and in http://forums.apc.com/message/14846#14846
it appears that a device plugged into the same electrical branch can dump electricity into the neutral that is coming out of my outlet, while http://forums.apc.com/message/2387#2387
is supportive, in http://forums.apc.com/message/16301#16301
there's a complicated answer I won't try parsing now, and I didn't check http://forums.apc.com/search.jspa?peopl ... +Fault%22*
for all the possibilities.
When I plugged the suppressor into another outlet elsewhere in the same apartment, thus likely on the same electric meter, the Building Wiring Fault LED no longer lit up. And (to correct a minor error in my posting above) the suppressor has an LED that signifies that protection is working and that LED is always on. So the suppressor is still suppressing and the problem is isolated to just part of the apartment. I wish I could trace the branch circuit to all the fixtures and outlets it connects but I don't think I have that kind of access.