We have been experiencing similar issues but only on one of our email addresses. Mailbox is definitely not full! I called the service centre twice - the first time I was told to change the password and that didn't work. Then I queued back about 90 minutes later (it was a long queue) and eventually I was told to use a different browser so moved from Chrome to Opera on the webmail (emails aren't coming into Outlook anyway)and the change of browser didn't work.
Lastly I was told I would have to wait for 36 to 48 hours for this to be fixed!! How rubbish is that?!
PS I have also deleted hundreds of emails and the 100% mailbox quota line hasn't budged a millimetre downwards!