The problem was (is) that application X requires Oracle 8, while application Y requires Oracle 7 and SqlNet. Ok, I'll install Oracle 7 in a separate location.
Danger! Danger Will Robinson!
Upon installing Oracle 7, I cannot get either application X or Y to work. Some tinkering with my PATH and TNS_ADMIN settings finally gets application Y working, but X is still broken - it can't see the Oracle database any more. I futzed around some more with the tnsnames.ora file, but nothing gets application X working again. This despite the fact that tnsping sees the database just fine.
I decided that application X was more important, so I tried uninstalling Oracle and reinstalling. For some reason now, I was only getting a partial install. I finally managed to find a full version of the Oracle installation program, copy it to my hard drive and re-install Oracle 8.
I even had desktop support pay me a visit and he had no clue. Why is Oracle installation such a PITA?