Generally caps fail in one of 2 ways....
1. Quietly, they get fat and just stop working...usually because of age or prolonged exposure to heat.
2. Loudly, with a bang and spray their electrolyte all over the rest of your board. This is unpleasant and smelly. Usually due to wrong polarity or over-voltage.
Electrolytics, plastic film and ceramic disc caps usually fail to open circuit and in most cases don't do any further damage to whatever kit they happen to be in at the time.
Tantalum caps 99.5 times out of 100 fail to a dead(ish) short and in some cases this can result in bad stuff happening to the rest of the circuit they are a part of.
It is a good idea to do a thought experiment about what happens in either case then decide on risking recycled parts, or something brand new.
For the ultra paranoid an ESR meter is essential for testing for deadness in capacitors.
Resistors usually last forever, if they get too hot they burn out, the smoke escapes and they stop conducting anything.
Before changing caps or resistors it is worth checking to see WHY they have failed. If the bit of kit is quite old and has been powered on for most of it's life, then it is usually safe to assume that all the electrolytic caps have dried out and will need to be replaced. This is almost worth doing as a matter of course after 10 or so years of heavy use. This rule may not apply in 10 years time.... Look to see if they're getting fat, this is a sign of old age and impending death in capacitors
Does this help?
C