I first started recapping antique radios in the mid 1980's and was a bench tech for a while in the 1990's. Below is from my experience, YVMV
1. Electrolytic capacitors are the most failure prone, many going off spec after just a few years, or even earlier due to the nature of their construction. Anything over 20 years old, they can and should be suspect. They have a moist electrolyte material that, no matter how well sealed, will dry out eventually. Many computers, flat screens, DVD players, audio gear, etc. suffered from the capacitor issue of the mid 2000's as pointed out above. This seemed to have been resolved before 2010. These are usually easy to spot though, as they usually vent when they fail. That being said, I've seen many electrolytic caps in 50's and 60's gear, usually in metal cans, that still test well, even under load. Don't know if I would trust them though.
2. In oil, and paper capacitors are the next most failure prone. Most all in oil caps I've seen, mostly from sixties gear, were bad. Like nearly 100%. Paper are not much better.
3. Ceramic and Mica capacitors have a much longer service life. I have yet to see a bad ceramic, although I'm sure there are some out there. Mica capacitors used to always be good, but I've ran across some in 1930-1950's sets that have gone bad. Wafer mica caps in IF and RF transformers are starting to become an issue. The caps are inside the transformers themselves. These were common in 50's and early 60's radio circuits. This is a fairly new occurance for me. 20 years ago, I almost never came across a bad mica but it's not uncommon anymore.
In newer gear, especially gear built after 1980, many manufacturers seemed to install capacitors that were just rated for the circuit they were in instead of having a little voltage cushion. So there really isn't a lot of wiggle room when a cap starts to go bad. Also some caps, especially electolytics, may test OK until a rated load is put on them, then they fall apart.