During an Exchange migration I found out that some of the mailboxes were listed as Lined Mailboxes, instead of User Mailboxes.
If you have these, you might have some issues giving certain permissions to those mailboxes.
To go around this issue I followed the process I found here.
So, to cut it short, the process is basically this:
Select the Linked Mailbox, right click it and Disable the mailbox. You will get a message indicating that the Exchange proprieties will be removed without removing the mailbox that you have to accept.
Then, open Exchange Management Shell and run the following command: