Yeah right, this is what you get if you have a smart-phone, tablet PC and laptop checking for new emails at the same time and from the same home network behind a NAT router.
Of cause only if you have Dovecot running with a default (possibly also upstream) Debian configuration, where only ten concurrent IMAP connections are allowed per IP and user. Setting the
mail_max_userip_connections to 20 solved the connection issues some of the devices where running into. Interestingly, according to the note above the parameter, the username is compared case-sensitively.