If I even remotely *think* I might want to check-out late, I ask for it when I check in. Since I only ask for 1 or 2PM, they almost always okay it right then and there. Most of the time, when I've called them from my room on the morning of check-out, I get the "we're full tonight" line and they insist that I get out by noon. I've noticed that my status hasn't had much affect here...
I get the sense that the front desk probably gets quite a few phone calls in the morning requesting late check-out. If they aren't in a good mood, they probably just start saying no to everyone.