Disclaimer ::: I am not a Plesk expert, use advice at your own risk ...
If you're have this issue, you will have the following in your boot.log ...
imapd.rc: bind: Address already in use
I'm not sure which one should be running courier-imap, daemon or xinetd, maybe someone will confirm ... my hunch and what I'm using is the daemon ...
If you have the following files in /etc/xinetd.d, move them to a safe place (actually I found a copy in /etc/xinetd.d/archive) ... this step will eliminate the possibility of imap/pop running from xinetd
/etc/xinetd.d/courier-imapd
/etc/xinetd.d/courier-imapds
/etc/xinetd.dcourier-pop3d
/etc/xinetd.d/courier-pop3ds
Next check that the following exists ...
/etc/init.d/courier-imap
... and is linked in /etc/rc5.d.
Then reboot ... check the boot.log for errors ... check Plesk CP services ... etc.
You can use the chkconfig command to list/add/delete boot entries ... my system looks like this ...
chkconfig --list
syslog 0

ff 1

ff 2

n 3

n 4

n 5

n 6

ff
winbind 0

ff 1

ff 2

ff 3

ff 4

ff 5

ff 6

ff
httpd 0

ff 1

ff 2

ff 3

n 4

ff 5

ff 6

ff
netfs 0

ff 1

ff 2

ff 3

ff 4

n 5

n 6

ff
network 0

ff 1

ff 2

n 3

n 4

n 5

n 6

ff
random 0

ff 1

ff 2

n 3

ff 4

n 5

n 6

ff
rawdevices 0

ff 1

ff 2

ff 3

ff 4

n 5

n 6

ff
saslauthd 0

ff 1

ff 2

ff 3

ff 4

ff 5

ff 6

ff
xinetd 0

ff 1

ff 2

ff 3

n 4

n 5

n 6

ff
named 0

ff 1

ff 2

n 3

n 4

n 5

n 6

ff
smb 0

ff 1

ff 2

ff 3

ff 4

ff 5

ff 6

ff
crond 0

ff 1

ff 2

n 3

n 4

n 5

n 6

ff
snmpd 0

ff 1

ff 2

ff 3

ff 4

ff 5

ff 6

ff
snmptrapd 0

ff 1

ff 2

ff 3

ff 4

ff 5

ff 6

ff
anacron 0

ff 1

ff 2

n 3

ff 4

n 5

n 6

ff
iptables 0

ff 1

ff 2

n 3

n 4

n 5

n 6

ff
portmap 0

ff 1

ff 2

ff 3

ff 4

n 5

n 6

ff
sshd 0

ff 1

ff 2

n 3

n 4

n 5

n 6

ff
courier-imap 0

ff 1

ff 2

ff 3

n 4

n 5

n 6

ff
qmail 0

ff 1

ff 2

n 3

n 4

n 5

n 6

ff
psa 0

ff 1

ff 2

ff 3

n 4

n 5

n 6

ff
mysqld 0

ff 1

ff 2

n 3

n 4

n 5

n 6

ff
ntpd 0

ff 1

ff 2

ff 3

ff 4

ff 5

ff 6

ff
mailman 0

ff 1

ff 2

n 3

n 4

n 5

n 6

ff
xinetd based services:
chargen-udp: off
rsync: off
chargen: off
daytime-udp: off
daytime: off
echo-udp: off
echo: off
services: off
servers: off
time-udp: off
time: off
time: off
finger: off
swat: off
ntalk: off
talk: off
smtp_psa: on
smtps_psa: on
poppassd_psa: on
ftp_psa: on
Good Luck !!!