A capacitor is made up of layers of metal separated by an insulator such as glass, mica or ceramic. One metal layer stores negative charges, another stores positive charges. The insulator layer keeps these charges separate from each other, so they can't leak away. However, capacitors can have defects that create a short circuit, which causes the stored electricity to leak across the barrier. This sudden electric current can cause enough heat to destroy the capacitor, damage surrounding components and even start a fire.
The Consumer Product Safety Commission conducted a study that showed that between 1990 and 1998, fans were responsible for approximately 4,500 electrical fires, 20 deaths, 270 injuries, and $55 Million in property damage. The study did not specify whether any of these cases were directly related to capacitor issues. It does, though, indicate that there may be good reason for concern.
T.C. Forensic reports that there is some evidence that fan fires were the result of bad capacitors. When fan capacitors go bad they can melt the plastic cases that they are housed in. A fan capacitor should be replaced if there is any sign of scorching or melting on the capacitor housing. The fan capacitor helps the motor start up and change speeds. If your fan starts experiencing problems starting or changing speeds, take a look at the motor for signs of scorching or melting plastic.
However, in 2004 T.C. Forensic concluded that fires caused by fan motors are very rare. This is because fan motors are well ventilated, and so dissipate heat very effectively. Even if the fan capacitor goes bad, it won't be able to build up heat very easily. The Consumer Product Safety Commission found that seized-up motors and accidents caused by user error made up a large percentage of the fire incidents. While it is possible for fan capacitors to cause a fire, they are not one of the more significant risks. It is much more important to make sure that the cord is not damaged by heavy furniture or sharp edges.