Summary: Among the most startling experiences a student encounters is learning that, unlike electrons and other elementary particles, photons have no mass. Under certain circumstances, however, the light quantum behaves as if it did have a finite mass. Starting from Maxwellâ€™s equations, we discuss how this arises when light interacts with a charged plasma, or travels along a waveguide. The motion of such photons is analyzed using kinematic concepts of special relativity, and we show how a cutoff frequency for effective propagation appears. Seeing how an environment may yield an apparent dynamic mass to the photon paves the way for later understanding: might the Higgs boson field provide other particles, such as the electron, with a mass? This paper is addressed to mid-level physics students, teachers and lecturers, requiring only a knowledge of classical electromagnetic and special relativity theories.