The broken hand is an injury in Far Cry 3.
Injury Obtained By...
This injury can pop up at random when healing without a syringe.
It is most likely to appear, however, when the player has fallen from a great height, for example off of a hill or out of a glider. The harsh crash to the ground is more likely to cause the thumb to bend backwards like it does when Jason catches himself from the fall.
The broken hand is healed by holding down the Q button to administer some DIY healing.