We investigate the resonance mechanisms for discrete breathers in finite-size Klein–Gordon lattices, when some harmonic of the breather frequency enters the linear phonon band. For soft on-site potentials, the second-harmonic resonances typically result in the appearance of solutions with non-zero tails, phonobreathers. However, these tails may be very weak, and for small systems where the phonon frequencies are sparsely distributed, we identify phantom breathers as being practically localized solutions, existing with frequencies in-between the phonon frequencies. For particular parameter values the tails completely vanish, and the phantom breathers decay exponentially over the whole system. We also describe briefly a first-harmonic resonance with a constant-amplitude wave and the generation of phonobreathers for hard potentials.