During import mailbox data from .PST file to a mailbox when you use Exchange Management Shell console, you can see this error:

Import-Mailbox : Error occurred in the step: Moving messages. Failed to copy messages to the destination mailbox store with error: MAPI or an unspecified service provider. ID no: 00000000-0000-00000000

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You should add mailbox permission to the user’s mailbox for administrator:
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Get-Mailbox -Identity albert.black@company.pl | Add-MailboxPermission -User Administrator -AccessRights FullAccess

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Useful links:

• How to Import Mailbox Data on Microsoft Technet: http://technet.microsoft.com/en-us/library/bb691363.aspx
• Get-Mailbox cmdlet on Microsoft Technet: http://technet.microsoft.com/en-us/library/bb123685.aspx
• Add-MailboxPermission cmdlet on Microsoft Technet: http://technet.microsoft.com/en-us/library/bb124097.aspx