How can I increase the number of Courier-IMAP connections for my Plesk server?

Symptoms

After mailbox synchronization process some folders are missing in Outlook.

Mail users see this error in an email client (like Thunderbird or Outlook or Horde or Roundcube) when accessing their mailboxes:

ERROR:
Unable to connect to your IMAP server. 
You may have exceeded the maximum number of connections to this server.

Or this one:

Error reading from network: cause: connection closed by foreign host

Or the following error in Horde and Rouncube:

Error when communicating with the server

Or, after a long delay:

failed login.

The following can be found in the server maillog (for example):

courier-pop3d: Maximum connection limit reached for ::ffff:10.0.0.1
If `dovecot` is used on server:

`mail_max_userip_connections` in the `Dovecot` configuration is set to 10.

The following error appears in `/var/log/maillog/`:

Dec 11 11:11:11 server11 dovecot: imap-login: Maximum number of connections from user+IP exceeded (mail_max_userip_connections=10): user=<example@example.com>, method=PLAIN, rip=11.11.11.11, lip=22.22.22.22, TLS, session=<NfqH3yFE4fAqhdEQ>

Cause

By default, Plesk and a Courier/Dovecot email server will limit the number of inbound connections to prevent users from opening up too many concurrent sessions. Unfortunately, this restriction can impact legitimate users who have multiple computers connecting to a Courier-IMAP/Courier-POP3/Dovecot server from behind a firewall, or a single computer running a mail client that takes advantage of mailbox-caching.

Resolution

• Go to Home -> Tools & Settings -> Mail Server Settings and set an appropriate value in Maximum number of connections per IP address

OR

For Courier:

• To remove this restriction and increase the permitted number of inbound connections, you can modify your IMAP/POP3 configuration file. The IMAP configuration is located at /etc/courier-imap/imapd , while the POP3 configuration is located at /etc/courier-imap/pop3d .

  1. Make a backup before changing any data:

     cp -a /etc/courier-imap/imapd{,.backup}
     cp -a /etc/courier-imap/pop3d{,.backup}
     

  2. Replace the following settings for IMAP:

     sed -i 's/^MAXDAEMONS=40/MAXDAEMONS=80/g' /etc/courier-imap/imapd
     sed -i 's/^MAXPERIP=4/MAXPERIP=40/g' /etc/courier-imap/imapd
       Replace the following settings for POP3:
     
     sed -i 's/^MAXDAEMONS=40/MAXDAEMONS=80/g' /etc/courier-imap/pop3d
     sed -i 's/^MAXPERIP=4/MAXPERIP=40/g' /etc/courier-imap/pop3d
     

  3. To apply the changes, restart the Courier-IMAP/Courier-POP3 service:

     /etc/init.d/courier-imapd restart
     /etc/init.d/courier-imaps restart
     /etc/init.d/courier-pop3d restart