Apparently this error isn't very common.
I find help for "Can't open the illustration" errors, but the errors described always contain more info. My error only says what I stated.
Illustrator gives the error, then "opens" the file, but it's blank.
And here's the kicker: I CAN open any of these files normally on another install of Illustrator on the network.

So it's not the files, though just for reference I tried .eps, and .ai.
And it's not an overall Illustrator error, because I have successfully opened some files.

Illustrator doesn't have a preferences file that I know of the way InDesign does, that I might throw away and make it rewrite.

Network permissions are fine, because the user can open/edit/save other files in the same drive and folder.

There's no version differences: we're all on CS1.

Restarting didn't help.

What else?