In Exchange 2007 and also most likely in Exchange 2010, you can give someone the full access permissions to a mailbox. But the person might not be able do send mails as this person. This happens because there’s a difference between the Send As and the Full Access permission. You might have the Full Mailbox permissions but as a matter of fact the Send As permission is not part of the Full Access permission. You need to give the user the right to send mails specifically.
All the information on the different permission types can be read in the Technet Article right here.