A quantum theory of high-order harmonic generation by a strong laser field in the presence of more bound states is formulated. The obtained numerical and analytical results for a two-state hydrogenlike atom model show that the harmonic spectrum consists of two parts: a usual single-state harmonic spectrum of odd harmonics having the energies $(2k+1)\omega$ and a resonant part with the peaks around the excitation energy $\Delta \omega$ . The energy of the harmonics in the resonant part of the spectrum is equal to $\Delta \omega \pm \omega ,\Delta \omega \pm 3\omega$ , …. For energies higher than the excitation energy, the resonant part forms a plateau, followed by a cutoff. The emission rate of the harmonics in this resonant plateau is many orders of magnitude higher than that of the harmonics generated in the presence of the ground state alone. The influence of the depletion of the initial states, as well as of the pulse shape and intensity, is analyzed.