I have on occasion remained in a room past the scheduled check out time. Don't leave the room, as your key will no longer work. But until Housekeeping comes to clean the room, at which point I apologize profusely and complain of not feeling well, I've never had an issue.
Having said that, most hotels have a late checkout if requested. We were able to stay in a Hilton hotel in Santiago, Chile until 3pm at no charge. If we wanted to stay until 8pm, it was a 1/2 day rate. Since many flights from Europe to the US are "red eyes", hotels are used to accommodating these requests.