Does an atoms mass number or atomic number ever change?
Yes and no.
An elements atomic number will never change, that is because the atomic number is its "identity." The atomic number is the number of protons that is contained in the nucleus, if you add a proton you change the element. Therefore, an elements atomic number will never change.
Atomic mass, however, can change and we call these isotopes. The atomic mass is the sum of the protons and neutrons in the nucleus, in an isotope you have more neutrons than you have protons making the mass larger than the stable nuclei.