If you think about capacitors, check out this video

No way to know what went bad on the board.  If you suspect certain caps, the only way to test this is to remove the cap from the circuit and test it with a meter that tests capacitance.  You could do a rough, in-circuit test with an ESR meter (equivalent series resistance).  For most home diagnostics, an ESR meter is more handy than a capacitance meter. 

I’ve had two boards from subwoofers go bad.  I’m wondering if the culprit is the metallized polypropylene film capacitors which are closest to the power source.  Looking at them, you can’t tell that those capacitors are bad, from what I’ve seen.  Any ideas?

If you suffered from that problem you would have known it by 2004. 

It wasn't so much a counterfeit capacitor as capacitors that used counterfeit electrolytic.  There's a whole story about a chemist who "stole" the electrolytic recipe and apparently didn't do it very well. 

None of this AFAIK has had any impact at all on audio gear.

A very young man  (just turned 18) who works as a part time tech at a local audio shop, has done repair/refurbishing of hundreds of pieces of gear.  He mentioned certain brands of caps that almost never go bad and others that he automatically replaces even if they look okay and have an acceptable ESR because he knows they will go bad in the future.  He particularly does not like certain Panasonic electrolytics (grey in color) because they almost always end up leaking and the electrolyte is very caustic and destroys board traces when they leak.  I just saw a Technics 1500 tape deck he worked on that he replaced 234 of these bad caps; it took him five hours (it would take me five years).  Most of the caps he pulled did show signs of just beginning to leak.  The owner of the deck is very lucky because there was no big leak damaging a board.

The good news is a failed capacitor almost always is detectable visually.  Open it up and inspect the caps.  If you find the bad one, it is pretty easy to replace.

Jerry