The fermion bag formulation helps to extend the concepts of topology and index theorem associated with non-Abelian gauge theories to simple lattice fermion field theories. Using this extension we can argue that fermion masses can arise either through the traditional mechanism where some lattice symmetry of the action that forbids fermion mass terms is explicitly, anomalously, or spontaneously broken, or through a non-traditional mechanism where all lattice symmetries continue to be preserved. We provide examples of simple fermion lattice field theories for each of these scenarios of fermion mass generation.