From detailed high-resolution measurements of the photon energy dependence of the (pi,0) superconducting-state photoemission spectrum of the bilayer Bi high-temperature superconductors, we show that the famous peak-dip-hump line shape is dominated by a superposition of spectral features originating from different electronic states which reside at different binding energies, but are each describable by essentially identical single-particle spectral functions. The previously identified bilayer-split CuO2 bands are the culprit: with the "superconducting" peak being due to the antibonding band, while the hump is mainly formed by its bonding bilayer-split counterpart.