Ample disk space with error: "1034 Incorrect key file for table 'table_name'; try to repair it"
Well, I'm a big dummy.  When I went to check the size of the database via the method requested by @a_vlad, I realized that previously I was checking the free space on /tmp on the wrong server.

Gah.  Executing the command again while tracking the free space on /tmp on the correct server reveals that lack of free space is, indeed, the problem.

...and here's the slice of humble pie I'm eating right now: