In the Thanksgiving episode where Chandler finds out that Monica cut off one of his toes, at the part where Monica is trying to cheer Chandler up by putting the turkey on her head, she puts the fez on the turkey. It's cocked right up on top of it i.e. vertically. When the camera cuts to Chandler and back to Monica, the fez is practically stuck to the back of the turkey. It's really obvious that they swapped it for a synthetic turkey with a fez sewn on to it and no, it didn't slip down, because if it had done that, it would have fallen off completely. It's obviously attached.