Think of what a cartridge is. You have a small block of plastic, wood, metal, etc. that is held in place on the tonearm by screws. There is nothing to wear out here. However, it is possible that the body of the cartridge could become deformed by environmental conditions over time or damaged during installation.
Inside the cartridge, there is a small magnet and a coil of wire. In a MM cartridge, the magnet is on the end of the cantilever. In an MC cartridge, the wire is on the end of the cantilever. Again, in the absence of physical damage, there is nothing to wear out. It is possible that that the magnet will weaken with time or that the wire coil will become corroded. To what extent this occurs, I don't know. There is another thing to consider here. The magnet or coil in the cartridge is often held in place by adhesives which could dry out and fail in time.
Next, a wire carries the signal to the pins at the end of the cartridge. Again, this wire could become corroded, but I don't know if this occurs, and to what extent it may affect the signal.
At this point, I do not see anything that could cause a properly constructed cartridge to fail. It should have a long and happy life measured in decades.
The cantilever/stylus assembly is another issue. The suspension for the cantilever is often a rubbery material which will dry out with time and affect the movement of the cantilever. But this is a stylus/cantilever issue, not a cartridge issue. And of course, the cantilever can become twisted and the stylus worn out. But again, this is not a cartridge issue.
In conclusion, I do not see how a cartridge will wear out or deteriorate other than over a very long period of time, unless there is shoddy manufacture or poor construction material, particularly regarding any adhesives. The cantilever/stylus is another matter entirely of course.
If somebody tells you that the cartridge is worn out, then that may be true for a particular cartridge. However, ask them what it is that is worn out, and why.