OK, here's some info that I wrote up and wanted to post, but figured it wasn't really all that necessary. But, since people are asking about *why* caps fail, I figured it might be interesting to some. It's definitely more info than you need - all you need to know is that caps fail with age and should be replaced, but if you want to know how they work, then here's a brief overview.
The secret life of electrolytic capacitors:
What is an electrolytic and why is it polarized? Why do these caps fail with age? Why is it they're usually blue? All good questions... well, except for that last one. That's just silly
First - the principle behind a capacitor. Two plates, separated by an insulator. A charge builds up on one plate, and is stored. Then, it discharges through the insulator to the other plate. In this way, a cap can smooth out a signal, block DC, and lots of other useful things. Normal capacitors, such as the ceramic disc types, have no polarity. Their plates are both the same. But, ceramic and plastic caps of this type have small capacities - less than 1uf usually. Electrolytic caps are special - they have polarity and higher capacities.
A typical electrolytic capacitor consists of two long thing strips of aluminum, sandwiched with a strip of paper soaked in a conductive electrolyte. The aluminum strips are the plates, and the insulator is actually a layer of aluminum oxide on one of them. The electrolyte soaked paper is not an insulator - but rather electrically becomes part of one of the plates. The leads are crimped into the aluminum foil, and everything is rolled up and stuffed into an aluminum canister, with a rubber plug sealing the bottom, allowing the leads to stick out.
The polarity of this type of capacitor is very important because of the thin aluminum oxide insulating layer. If you power the cap backwards, the reverse voltage will break down and destroy the insulating oxide layer, causing a short. This causes the cap to heat up and sometimes explode.
Modern electrolytics are provided with a scored vent in the top. This is to allow a "safe" release of pressure should they short or overheat. The vent will bend and split and release the gasses formed by the broken-down electrolyte. Occasionally you'll see a cap bulged up at this vent, or split. This is an obvious indicator of a damaged capacitor that must be replaced. Some older electrolytics have no such vent, or a different sort of vent. These usually vent out the bottom, or simply explode. It's also not uncommon to have a cap that has pushed the rubber plug partway out of the bottom.
That's all well and good - why do caps fail with age?:
Ah... that all has to do with that electrolyte solution. It's exact composition varies between manufacturers, but it's basically some kind of weak acid or solvent, usually mixed with ethylene glycol and water. In other words: oily goo. The capacitor's body is supposed to be sealed, but the electrolyte is rather corrosive in and of itself, and the main seal in the cap is a rubber plug crimped into an aluminum tube. Over time, this leads to the electrolyte evaporating, breaking down, and otherwise going away. The less electrolyte, the poorer one of the plates conducts, and thus the "capacity" of the capacitor decreases. In circuit, it no longer performs as it did when it was new. Also, because the plate doesn't conduct like it should, the effective resistance through the cap increases, and thus the voltage drop across the cap increases. Not good.
So, I can just replace the ones that are bad?
You could. Just so long as you like working on the same monitor over and over again. Some people like to use an ESR meter to find the caps that have started to fail or have failed - checking to see which ones have high effective series resistance. This will identify the bad caps, and then you can replace just those, and the monitor will work again. But for how long? Who knows. You could have gotten a couple of exceptionally good caps that'll last forever. But chances are, the rest of the 25 year old capacitors aren't far behind in their quest to dry out their electrolyte, and now that the monitor is running again, they'll sit there, slowly cooking away... They might last a week, a year, forever? Who knows. But the simple fact remains that electrolytic capacitors DO have a service life, and they do fail with age. Might as well just replace all of them while you're in there - the small ones in monitors are not expensive.
That means that capacitors have a shelf life?
They sure do. The manufacturer generally specs ten years of shelf life, or something like that. Effective service life seems to be something like 25 years... but it really depends on the application. Sure, a capacitor that's been sitting doing nothing for 20 years is more likely to work than one that's been stuffed in a hot flyback cage in a running game for the same length of time... but a brand new component is likely to last much longer.
-Ian