The grace period specifies how long the mailbox can be over the quota (disk space or number of messages) before messages are bounced back to sender. Messages are accepted by the MTA, but remain in the MTA queue and are not delivered to the message store until one of the following occurs:
The mailbox no longer exceeds the quota, at which time messages are delivered to the mailbox.
The user has remained over quota longer than the specified grace period, at which time the server bounces all messages including those in the queue. This time limit is controlled by the quotagraceperiod configutil parameter.
The message has remained in the message queue longer than the maximum message queue time. This is controlled by the notices MTA channel keyword (see To Set Notification Message Delivery Intervals).
For example, if your grace period is set for two days, and you exceed quota for one day, new messages continue to be received and held in the message queue, and delivery attempts continue. After the second day, the messages bounce back to the sender.
The grace period is not how long the message is held in the message queue, it’s how long the mailbox is over quota before all incoming messages, including those in the message queue, are bounced. The grace period starts when the user has reached the quota threshold. See To Specify a Quota Threshold and been warned.