Thanks, Dan and Don, the wonder twins. Dan's suggestion of using "lsof" (a new util to me, tho prolly not for smarter people). I installed it from the ports tree, and lo-and-behold, I think I've got a culprit, or at least a couple of suspects.
The box I'm running all this on is a poor benighted Dual Pentium Pro, that's running development for all of our web projects, as well as a forwarding DNS, Samba services, AppleTalk services, and a partridge in a pear tree. Shameful, I know...
The two biggest open file abusers appear to be postgres and apache -- I'll keep watching it, and will probably write some poor excuse for a perl script to watch it and see who's being naughty or nice.
I'm also going to try your advice, Don, and install a vanilla AOLserver and hammer it. I think, though, it's more likely a postgres problem. I am running the last beta release of 7.0, so I'll try the release version as well.
If I'm forced to move to Linux, I might cry, though. I sure do like my FreeBSD... :)