Semiconductors are made by adding doping substances to a slice of single crystal silicon. Dopants are atoms with 3 or 5 electrons in the outermost electron shell. This creates, in some regions, extra electron and, in others, extra "hole" (the lack of an electron - a positive charge carrier). The semiconductor effect is obtained by controlling when - and how many - these charge carriers can move freely. The C-MOS technology uses an electric field to allow the charge carrier to move freely - a capacitor with one plate being the metal contact and the other the region called "channel" where the electrons will move from the "source" to the "drain". The numbers in nano-meters mentioned for CPUs are the width of the channel.
The above paragraph seems complicated but is needed to explain something called "electron migration". So, how do we add boron or phosphor to a silicon crystal? Originally they used ovens to heat the crystal and that allowed the dopants penetrate the crystal. Controlling the temperature and time it is possible to create layers with different dopants. These dopants will continue to move inside the crystal but very slowly. The problem is that the speed that they move increases exponentially with temperature.
All this is to explain why semiconductors will always fail eventually and why keeping things cool is important. "Electron Migration" is the name given to the movement of these dopants. Chips can have billions of transistors (though router chips have very fewer) and, therefore, the failure is not sudden - some of the transistors will start to operate out of spec and then some will start to fail. Chips can be so complex that they will continue, in some cases, to function eve with some non-working transistors.
I Don't know the specifics relative to routers but, in general, this mechanism of failure applies to them, also. There are other causes for failure like @John mentioned.