The jumping cholla, Cylindropuntia fulgida, is also called the "hanging chain cholla" and is usually found in the southwest portion of the United States and northern Mexico. This cactus has tiny stems with spikes and eject off the plant when you get too close. It gives the illusion that the tip has jumped onto the person and attached itself to either their skin or clothes. Many of the desert critters can be seen running around with the cactus attached to their bodies.
I have not been attacked by one of these cacti but know several people who have. It is not easy getting the spiny stems out of your skin and it can be very painful.