Capacitors definitely go bad. You can source replacements by finding capacitors with the same uF rating online. The voltage rating can be (and is OFTEN) higher, but it cannot be lower. So for instance if your cap says 10,000uF 16v, you can substitute this with one that is 10,000uF 100v. These are exact matches besides the current it can handle. Obviously "100v" can handle your "16v". But you CANNOT substitute the uF value like this. (Read below)
Sometimes on the really old stuff they don't make that value (uF) anymore and that's when you need to start sourcing a similar replacement, but then the game plays a different speed or something else is messed up, etc., and it's really a pain it seems so hopefully you can get direct replacements. I would wait for somebody else to chime in as well. Sometimes there's just those capacitors that never seem to go bad and it's some other part that goes bad or is out of wack.