Scenario: You have arbitration mailboxes that are located on databases that are no longer accessible. Although the mailbox is no longer accessible, the AD User account is still in tact.
Solution: Disable and re-enable the arbitration mailboxes. The Database DB04 is where the arbitration mailboxes lived but DB04 is gone.
Set-adserversettings -viewentireforest:$true
$1 = get-mailbox -arbitration | Where Database -like DB04
$1 | Disable-mailbox -confirm:$false
$1 | Enable-mailbox