With such small & fragile bones, breaking a limb happens more often than you'd think.
broken rear limb
The largest cause of broken limbs is having that limb catch through wire (such as an uncovered wire ramp or a chain in a toy) and then the chinchilla breaking the limb while trying to get out.
There are 2 classifications for broken limbs... a simple break, in which the bone is broke but there is no outside wound, and a compound fracture, where the bone has broken through the skin allowing bacteria to get in.
A compound Fracture
As the saying goes "An ounce of prevention is worth a pound of cure", a few simple changes to their environment can greatly reduce the risk. Chinchillas do not need ramps generally-speaking, being very good jumpers, but if you have ramps in their cage, be sure they are covered in fleece so their limbs can't poke through the wire rungs. Any hanging toys should have the chain replaced with a safe alternative such as seagrass rope or solid wire, to remove the chance of tiny chinchilla hands or feet catching in the links of the chain.
A fracture of either kind will require veterinary care. Fractures can sometimes be healed by splinting the broken bone and limiting movement while the bone heals. More complex breaks, such as we see here with Ozzy (owned by Emily Grace Farrell) may require amputating the broken limb. When they amputate, they will take the whole limb as leaving a stump is a bad idea as they'll continually break it open. Please see our Wounds page for further reading.
Ozzy got caught between a wire, fell and twisted the leg - hanging upside down for god knows how long. He chewed through the muscle, nerves and bones to try and free himself. This was on a Savic Royal Suite cage.
Amputees, or what we affectionately refer to as tripods, do adjust and can go on with a high quality of life. One thing Emily shared is if you have a male tripod, you must do regular hair ring checks. Without a rear limb, your male chinchilla will be unable to sit on his haunches and clear any rings himself. While a missing front limb will also make it impossible to clear himself. Also, as with any open wounds, dust baths should not be given until the skin has closed up completely.
In your tripod's cage, you'll want to add some extra ledges, hammocks and such. Make sure that if he were to miss a jump, he wouldnt fall far without landing on something else.
In this video, 4 weeks after Ozzy's amputation, you can see he's regained full movement allowing him to bath, run and jump normally.