Capacitors (similar to light bulbs) will fail at some point. Sometimes the chemicals (an electrolyte/oil/etc..., though PCB are very stable) will degrade and cause the capacitor to expand (perhaps blow up). Other times, the dielectric material (the material that separates the two conductors in the capacitor) will degrade, allowing the terminals of the capacitor to short together (perhaps causing the capacitor to blow up). Many air conditioner capacitors have a "feature" that causes the capacitor to become disconnected when they expand, reducing the chance of explosions.
The lifetime of the capacitor is greatly impacted by temperature and voltage. At higher temperature and higher voltage, the capacitors will fail more quickly. The temperature can be influenced by dust and debris (insulating the capacitor) in the compressor unit. The voltage should be OK, except for transients like lightning strikes. Also, keep in mind that that dirty coils/improper charges will cause the system to be less efficient and run for a longer amount of time each day. This will cause the capacitor fail at an earlier date since it's being used more heavily.
Air conditioners often will have two capacitors: "start" and "run". These have slightly different requirements, but for the purposes of this post can be considered to be equivalent.
There are two main technologies used for the creation of capacitors for air conditioners: electrolytic and polypropylene film.
Electrolytic capacitors are more sensitive to temperature and have a propensity to expand, causing premature failure. Polypropylene capacitors generally are more temperature-resistant AND can self-heal. When polypropylene capacitors have an internal short, they tend to just vaporize the local area, leaving the rest of the capacitor to function normally.
Some datasheets of polypropylene capacitors I found state that 90-94% of the capacitors will last >=60,000 hours. Electrolytic capacitors are often used as "start" capacitors and are only used for a short time (on the order of a second) each time the AC unit turns on. They are often rated for on the order of 40,000 starts.
So, if you want a longer life capacitor in the future, keep your air conditioner unit clean and well maintained, and use polypropelyene capacitors, if possible. Also, you could install a whole-house surge suppressor in order to reduce power supply surges.