Of course a few seconds later, the pins and needles start. The answer is no. The whole effect is caused because your nerve cells have stopped working. When you apply pressure to your arm or foot, your body's cells become starved for oxygen because the flow of blood is reduced. All cells are affected, but the cells that are the most effected are your nerve cells.
Nerve cells come in many types.
Both types of nerve cells have coatings surrounding them called myelin. However, motor nerves have thicker and more sensitive myelin layers. Now let's get back to the subject of sleeping feet. When a nerve cell is starved of oxygen, it stops working. Motor cells, with their thick myelin layers, are the first to stop working.