In electronics, electrolytic capacitors are the first component to fail, and those are frequently used to filter ripple out of DC rectifiers on power boards. A lot of places treat larger capacitors as having a 5-10 year lifespan before they need replacing. If a circuit board is being refurbished, usually all the electrolytic caps are replaced.
The main way to identify a failing electrolytic is to see if the top is bulging or leaking- that's from the cap shorting out, heating up, and causing the electrolyte to vaporize. If the caps are on a power board or did not destroy other components by failing, replacing these caps will usually be enough to revive the device.
Another failure point is older solder joints. As the joint flexes over time (either from age, movement, or heat expansion), the joints get brittle, break, or become non-conductive. This seems to be particularly common with lead-free solder joints, which started being used in the 1990s. Reflowing the joint is usually enough to restore a connection, and may be all that's wrong with some logic boards. Some vintage computer enthusiasts have revived old computers by "baking" the motherboard in the oven for an hour.