The overwhelming majority of tickets are non-refundable. I think that the rule you cite is referring to non-refundable tickets. I have purchased refundable tickets, checked in, not flown, and gotten a refund. (I have also asked for a refund at the airport and been told that my ticket was non-refundable -- by an agent who did not even look at my reservation. When I pointed out that it was I was told to just call in for my refund)

You can also reverse the check-in process of non-refundable tickets prior to flight by calling CO. They don't like to do it, but they will if necessary.