The most prone capacitors to fail in vintage radios and electronics are:
The Electrolytic capacitor: is used in the power supply of radios and other equipment to filter out the rectified dc voltage from the power transformer. It is made of two aluminum sheets separated by a paper soaked electrolyte. Oxide forming occurs at the factory with low current variable voltage. The aluminum oxide layer is the dielectric insulator between the plates. Electrolytic capacitors are wound in such a way that extremely large values of capacitance can be achieved.
Failure Modes
The electrolytic capacitor can fail open, if the oxide layer has deteriorated and the electrolyte has dried out. The characteristic will be a very loud hum from the speaker no matter the position of the volume control.
The electrolytic capacitor can fail short circuited.
If the capacitor has sat idle with no voltage applied for months or years, the oxide layer will deteriorate and will have to be regenerated by a low current voltage increase over a period of days.
If you plug in your old radio and turn it on, the capacitor always represents a brief short circuit before it charges completely. However, if the oxide layer has deteriorated, the initial inrush of current can cause the capacitor to permanently short circuit, destroying power resistors, filter chokes, speaker field coils, rectifier tubes and solid state diodes, and lastly the power transformer itself.
The electrolytic capacitors have a life span of 20-30 years for the high quality brands. There are two main brands of electrolytic capacitors used in my shop: Nichicon and Panasonic, with an upgraded temperature of 105 deg c.
The plastic film capacitor: is used to couple audio signals and bypass audio and radio frequency from entering the power supply. It is made up of either foil sheets separated by paper, or metal coated plastic sheets and the plastic or paper is the dielectric insulator. The types of film capacitors found in vintage radio, audio and electronic gear, is most commonly referred to as the wax paper type. This type of film capacitor is metal film with paper and is sealed in bees wax. The bees wax was thought to seal out moisture, but it is not a very good sealant, allowing moisture to deteriorate the paper. All modern film capacitors are sealed with epoxy that is actually moisture resistant. Some film capacitors are metal paper and enclosed in plastic cases.
Failure Modes
The problem with the metal & paper film capacitor is the deterioration of the paper insulator, causing them to fail shorted. The deterioration of the paper causes the film capacitor to leak electrically. Instead of blocking dc voltage and passing ac voltage, it passes ac voltage along with dc voltage. On most tube gear this causes a bias problem on the grids of tubes and can completely silence a radio and cause red plating of the power tubes in radios and amplifiers. The capacitors themselves can fail and turn into a resistor, which will cause them to overheat and melt the wax and more. They will eventually short completely and I have seen some destroy other components. I have seen plastic cased metal paper capacitors, split open, arc, smoke etc.