I poked around on the 3com BBS tonight, and I found a little note detailing
known problems with the 3c509 not resetting correctly on EISA machines after
a warm boot (our machine is an EISA machine).  3com's solution is to put
your card in EISA mode, and setup the card with your EISA config utility.
I did this and now the card probes perfect every time!  I'm not sure if this
helps anyone else who's having this problem, but it might.

Anyway, thanks for all the suggestions that people sent in!

--Ken