After setting up the backups I changed the database "Compatibility level" from 80 to 90. The nightly backup ran fine that night, but the transaction log backups failed consistently. Turns out that changing the database compatibility level wipes out the database's knowledge of when it was last backed up, and a transaction log backup will only run after a full backup. So I ran another full backup, and then transaction log backups started to run just fine.
The only useful error message, that pointed me to the real problem, was in the SQL-Server Integration Services Output file. The History log, the SQL-Server logs, and the server Event logs were pretty useless. So, always set up job step output files for every step, even if it seems unnecessary.
