In my experience - it is basically up to the agent on the other end of the phone. From what I understand of the computer system, there is no computer 'lockout' preventing a passenger from being upgraded from any fare at any point in time. If there were, then it would be near impossible to upgrade companions who are not frequent fliers, and extraordinarily complicated for CP passengers using the old stickers. If they say that 'the computer won't do it' they are really just saying they won't do it, and that they want you to call back later.
There is no waiting list for upgrades although other carriers have one. It would make all of our lives easier if there were...