If your server mailbox is full, you will need to clear space by removing messages from the server. If senders report messages being bounced back to them with an error, "Mailbox is full/blocks limit exceeded / inode limit exceeded," Please perform the following steps in Outlook.
These directions work for Windows 10 and the latest version of Outlook.
Go to your Outlook settings using THIS METHOD.
Once there, click the Advanced tab and make sure messages are being removed from the server during a reasonable time frame. You should also check the "Remove from server when deleted from 'Deleted Items; box. Click OK to save the settings, and so on. If this doesn't work for you, you may use webmail to delete unwanted messages.
This method only works with a POP account. IMAP leaves messages on the server by default. If you delete messages using an IMAP client, they will disappear from all devices and be deleted from the server!