Hi @Markus Walter,
After i restart both onpremise Server i can login with IMAP to this Mailbox. Then it takes a while and it isn´t possible. It seems that it is specific to this Mailbox.I did a test with a new created Mailbox there isn´t an issue.
Are you testing in the same network? If not I would suggest switching to another network to see if it is a network issue. If this is not the case, please run the following cmdlet in Exchange Management Shell on the Exchange servers to check if it will return error messages:
Test-ImapConnectivity -ClientAccessServer Contoso12 -MailboxCredential (Get-Credential contoso\kweku)
Besides, please also have a check in application log on Exchange servers to see if you can find error or warning events which may be related.
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