I am in the camp of not changing every last capacitor in tv's and old radios.... As a early tv tech this is what I can say about caps.... Old paper/foil & Wax caps were the best of the time.... High voltage placed across two large rolls of foil insulated by an oily wax paper will eventually break down, dry out, and allow current to flow from one layer of foil to the next..... Problem for testing is that current may only flow at 100Volts or more..... 50 volts or more.... 200v or more..... Are they better now.... In some respects yes.... But circuits are not the same today..... Some design parameters are different, and what was common in 1966, may be a harder to find item today....
Now some caps like the big filter caps in the power supply have a nice large current supply available, so they may get hot, they may boil, and then explode..... They are the most common caps that will go bad, and most older ones made in 1959 or earlier should probably be changed for safety....
Other smaller caps can be fine, or may become leaky.... Some specific types, or brands may have a history of being bad..... Some companies may have had a habit of using lower quality stuff, like RCA, there is a lot written here about how a lot of zenith stuff outlasts rca's stuff in the cap, and picture tube departments....
Cap testing falls into two parts. Testing for value, and testing for breakdown at working, or rated working voltage. For testing at working voltage you need a tester like this one:
http://www.ebay.com/itm/Heathkit-IT-...item4d108c603f
You need an old one, where the tester has the ability to put 300v across a cap rated at 250v and be sure it is not going to break down....
ESR testers are useful for power supply filter caps as well, and can tell if you are going to have a potential bad cap in the future.....
http://www.ebay.com/itm/New-Capacito...item258174ca27
Wikipedia has a little on the topic of esr's in caps...
http://en.wikipedia.org/wiki/Electrolytic_capacitor
Why not change all caps just to do it...? Caps have a lot of parameters that have to be matched for the circuit they are going into. For old radios and tv's temperature stability is important, high voltage rating too. Have you looked at all the factors to pick a good choice to replace every cap in your tv ? Are you that good an engineer to do that ? Are you that good at tv work not to make a mistake in replacing them ? Are you sure the new cap will not immediately introduce a problem you have not already got that will be almost impossible to find because you "know that new cap is good"? Are you replacing all the caps while the tv or radio has a problem you have not fixed first.... ? This is a sure way to give yourself your original problem plus new ones.... Look at all the problem threads here that begin with "I got this picture rolling problem after recapping my set...."
There are lots of notes here about known bad cap types, and caps used in specific circuit areas that should be changed, others are most likely best left alone....