We observe a crossover from electron-phonon (e-ph) coupling limited energy relaxation to that governed by thermal boundary resistance (phonon-phonon coupling, ph-ph) in copper films at subkelvin temperatures. Our measurement yields a quantitative picture of heat currents, in terms of temperature dependences and magnitudes, in both e-ph and pp limited regimes, respectively. We show that by adding a third layer in between the copper film and the substrate, the thermal boundary resistance is increased fourfold, consistent with an assumed series connection of thermal resistances.