We've gotten it every time we've asked so far.
When we fly up, our flight home is also late - the Magic Express comes to get us around 6PM for I think a 9:30PM-ish flight. So the later we can stay in the hotel in the morning, the better, before being "stranded" all day.
I found out you don't ask at the front desk. You need to call Mousekeeping as soon as they open (which if I remember correctly is around 6AM.) Ask if late checkout is possible. Typically they have had no problem giving us until 1PM, but one time it was only until noon. I don't think they do later than 1PM. If you ask too late, you're not going to get it. If you ask the front desk, the front desk is going to ask Mousekeeping.
We then leave our bags with bell services, go to the park(s) and then typically schedule a 5PM meal at the resort restaurant, so by the time we are done with dinner (which is nice and relaxing and of course there is a restroom) we can just walk outside to get our bags, pick up the Magic Express and get back to the airport.
Also, the one time we stayed at CBR, since the actual room is nowhere near the checkout building, we called the front desk to ask where to leave our bags, and within 5 minutes they sent someone to pick them up right from our room and take them to where we could pick them up later.