I noticed my mail server was getting some service temporarily unavailable errors and wasn't able to deliver emails, so after digging into it I found the cause was Postfix was unable to connect to Amavis.
When I looked at Amavis' logging (which I found in /var/log/mail.warn), I found this:
[snip] (!!)TROUBLE in pre_loop_hook: db_init: BDB no dbS: BDB0002 __fop_file_setup: Retry limit (100) exceeded, File exists. at (eval 91) line 318. [snip] (!)_DIE: Suicide () TROUBLE in pre_loop_hook: db_init: BDB no dbS: BDB0002 __fop_file_setup: Retry limit (100) exceeded, File exists. at (eval 91) line 318. Why is this happening, and how can I fix it?
I'm running Debian, and I believe I had both updated packages and rebooted shortly before this started happening, so maybe an update broke something.