Continuity mistake: After JD has done the "Eliott got some booty" song they both walk through a door, a patient in a red dressing gown then walks past. He wasn't there before they started walking and wouldn't have had been able to get there in that amount of time.
Continuity mistake: As Dr Cox is talking to J.D. towards the end of the episode and telling him that he won't let him treat the patient he traded, Dr Cox is putting on his white overcoat. If you look at his collar on the right side it is a little crumpled and not quite straight. The next shot we see over his shoulder and the collar has straightened out. The following shot it returns back to being crumpled up a bit.
Continuity mistake: A patient in a red dressing gown is walking around the background when JD is teasing Eliott. He is seen twice but wouldn't have been in the second shot with the direction he was walking and in the time taken for the shot.
Continuity mistake: The letter that JD drops on to Turk's tummy changes angle between shots.
Answer: No, it was a fake, they used a stamp.
It is real.