By default Plesk IMAP maximum connections per IP is set to 4. If you are using an email client that checks multiple mailboxes, this limit is easily reached.
To increase the limit, edit "/etc/courier-imap/imapd":
MAXDAEMONS=80
MAXPERIP=40| Plesk imap connections limit
            Submitted by sandip on Wed, 01/19/2011 - 16:49
    
     By default Plesk IMAP maximum connections per IP is set to 4. If you are using an email client that checks multiple mailboxes, this limit is easily reached. To increase the limit, edit "/etc/courier-imap/imapd": MAXDAEMONS=80»  
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cpanel courier-imap config location
The same limits can be applied for cPanel running couriertcpd.
The location of the config file is at:
/usr/lib/courier-imap/etc/imapdRestart courier-imap after config changes via:
/etc/init.d/courier-imap restart