Issue: 

User cannot access mailbox & sending email to this type of mailbox will receive an NDR stating “Your message wasn't delivered because the recipient's mailbox is quarantined”

Cause:

The mailbox is a potential threat to the health of the Information store and has been quarantined. There are several types of events for which the Exchange store tags a mailbox as a potential threat:
  • If a thread doing work for that mailbox fails
  • If there are more than five threads in that mailbox that haven't made progress for a long time
Detail:

This is called Poison mailbox issue, any mailbox identified as potential threat to the mailbox database will be quarantined by exchange servers, since Office 365 is a Multi tenant environment we cannot do a manual process to fix this. However by default exchange resets Quarantined mailboxes in 24 hrs after the quarantine flag is set.

How to Find whether the mailbox is in Quarantine, Run below command to retrieve Quarantine status of the mailbox:

Get-MailboxStatistics UserName | fl *Quarantine*

IsQuarantined : True
QuarantineDescription :
QuarantineLastCrash :
QuarantineEnd : 6/18/2015 3:57:37 PM
QuarantineFileVersion :

Reference Poison Mailbox: https://technet.microsoft.com/library/bb331958.aspx