Your archive mailbox is almost full

1. Connect to Exchange Online Powershell.
2. Run the following commandlet for the user: Enable-Mailbox <user mailbox> -AutoExpandingArchive
3. Run the following commandlet to confirm it is enabled for the user: Get-Mailbox <user mailbox> | FL AutoExpandingArchiveEnabled

 

As per below example:

Connect-ExchangeOnline

then run the following command to confirm the current "AutoExpandingArchiveEnabled" status. if it is not enabled/false. 

Get-Mailbox nawaz@domain.com| FL AutoExpandingArchiveEnabled

AutoExpandingArchiveEnabled : False

Lets enable it true with the following command.

PS C:\Users\Nawaz> Enable-Mailbox nawaz@domain.com -AutoExpandingArchive

Name                      Alias           Database                       ProhibitSendQuota    ExternalDirectoryObjectId

----                      -----           --------                       -----------------    -------------------------

nawaz     EURPdfR07DG198-db121             99 GB (106,300,44... 788cddcf3-f92f-4497-b95c-2b1dcea6982dff

And lets check it again.

PS C:\Users\Nawaz> Get-Mailbox nawaz@domain.com | FL AutoExpandingArchiveEnabled

AutoExpandingArchiveEnabled : True

To force the MFA (Managed Folder Assistant) to immediately process a specified mailbox

run 

Start-ManagedFolderAssistant -Identity nawaz@domain.com