Sickness
Pinatas in the Xbox 360 game never die. If you hit a pinata with the shovel, they can become ill, or even break open. Even when broken open, pinatas will reappear at the outside edge of your garden.
If a pinata does not have enough room to roam, they become ill. It's up to you to ensure that each pinata has adequate space to move about. A pinata can also randomly get sick, even with enough room to roam.
Should this happen, you can choose to ignore the pinata's ailment, or attempt to cure it, by using a bottle of medicine or hiring a friendly helper named Dr. Patchingo, who will heal your ill piñata for you. There is also a Witch Doctor named Dastardos who probably isn't very helpful. He'll try to break open your sick piñata to put it out of its misery. Should a pinata get too sick to heal, it will leave your garden and return to the wild. It will also discard any accessories you have put on it.