"Your P.M. boxes is full, and you could not read messages until it is cleaned out."

But I cannot see any button for deleting messages. What am I missing?

when you click on the messages and have the full list . click all then the button says something odd at bottom of them . anyway that's the delete button. no idea why it doesn't say delete.

hope that helps.

I still don't get it

I click the box on all the messages that I want to delete, click on them, but it brings me again to the page with the "Your P.M. boxes is full, and you could not read messages until it is cleaned out." warning. No button to delete

Edit:
I've found it!! you have to click on the "Submit" button

Why submit? Submit to the purge!

it says submit at the bottom I just checked . click all and submit! or when reading an individual message just below it also says delete.