Much like in humans, chinchillas can fracture (or break) bones. Luckily, chinchillas heal quickly from fractures. However, they require sufficient rest and proper restraint during recovery, so as not to aggravate the injury.
A chinchilla suffering from a fracture will suffer from immense pain, be unable to move -- especially the affected portion of its body -- and have swelling around the area of the broken bone. A crackling sound may be heard when the fractured area is manipulated due to the rubbing between the broken ends of the bone. Very rarely do chinchillas have an open wound on its skin. But if this occurs, you will be able to see the broken end of the bone piercing out.
Fractures can occur due to accidents, such as improper handling or the trapping of a chinchilla's legs in the cage's wire mesh, or because of nutritional disorders like calcium and phosphorus imbalances -- wherein the bone becomes brittle and is prone to break.
Your veterinarian can easily diagnose a fracture in the chinchilla by observing its clinical symptoms. Occasionally, however, they will confirm the diagnosis by taking an X-ray of the affected area.