To increase this restriction, modify
/etc/courier-imap/imapd
and change MAXDAEMONS and MAXPERIP to a more sane number.
In the case of my configuration, I changed MAXDAEMONS from 40 to 80 and MAXPERIP from 4 to 40.
This allows all the machines behind my home firewall to connect to multiple accounts on the e-mail server with mailbox caching enabled.
restart the courier server with:
/etc/init.d/courier-imapd restart
This has fixed Mac Mail and my iPhone, which were giving connection errors on a regular basis. Thanks!