In my case I was lucky and saw event 10014 from MSExchangeIS Mailbox Store saying "Available disk space for the database logs is extremely low. Message delivery to the database may be stopped.". There was still 1GB free so inbound email was still being queued on the Exchange server but not delivered to any mailboxes.
There were also many reports of event 629 from ESE "While attempting to move to the next or previous node in a B-Tree, the database engine skipped over 10421 non-visible nodes in 42 pages."
I traced the cause back 4 days to event 4121 from MSExchangeRepl saying "The Microsoft Exchange Replication service encountered an error when retrieving the database copy statuses during the periodic operation 'AmPeriodicDatabaseStateAnalyzer'. The operation will be retried later." When replication fails the transaction logs are blocked from flushing after full backup. This caused a buildup of 90GB worth of transaction logs which needed to be deleted.