There were a few warning popping up in my /var/log/mail.warn log for Postfix, like this:

Feb  4 09:16:15 sixsigma postfix/tlsmgr[3394]: warning: request to update table
btree:/var/run/smtpd_tls_session_cache in non-postfix directory /var/run
Feb  4 09:16:15 sixsigma postfix/tlsmgr[3394]: warning: redirecting the request
to postfix-owned data_directory /var/lib/postfix
Feb  4 09:16:15 sixsigma postfix/tlsmgr[3394]: warning: request to update table
btree:/var/run/smtp_tls_session_cache in non-postfix directory /var/run
Feb  4 09:16:15 sixsigma postfix/tlsmgr[3394]: warning: redirecting the request
to postfix-owned data_directory /var/lib/postfix

I fixed the problem by reconfiguring /etc/postfix/main.cf and changing this:

smtpd_tls_session_cache_database = btree:/var/run/smtpd_tls_session_cache
smtp_tls_session_cache_database = btree:/var/run/smtp_tls_session_cache

To this:
smtpd_tls_session_cache_database = btree:/var/lib/postfix/smtpd_tls_session_cache
smtp_tls_session_cache_database = btree:/var/lib/postfix/smtp_tls_session_cache