Recently had a user not be able to delete items from another Users exchange Mailbox ( 2010 )  apart from having full permissions to the Mailbox 

Exchange 2010 Mailbox has a 30GB limit of the Recoverable Items Folder

To find this current quota run : 

Get-MailboxFolderStatistics -Identity %username% -FolderScope RecoverableItems | Format-List

To delete them , run : 

Search-mailbox -identity %username% -SearchDumpsterOnly -DeleteContent

If you can’t find Search-Mailbox , you need to assign the below role to the user and restart exchange shell

New-ManagementRoleAssignment –Role "Mailbox Import Export" –user "administrator"