A comprehensive inelastic neutron scattering study of magnetic excitations in the near optimally doped high-temperature superconductor YBa(2)Cu(3)O(6.85) is presented. The spin correlations in the normal state are commensurate with the crystal lattice, and the intensity is peaked around the wave vector characterizing the antiferromagnetic state of the insulating precursor, YBa(2)Cu(3)O(6). Profound modifications of the spin excitation spectrum appear abruptly below the superconducting transition temperature T(c), where a commensurate resonant mode and a set of weaker incommensurate peaks develop. The data are consistent with models that are based on an underlying two-dimensional Fermi surface, predicting a continuous, downward dispersion relation connecting the resonant mode and the incommensurate excitations. The magnetic incommensurability in the YBa(2)Cu(3)O(6+)(x) system is thus not simply related to that of another high-temperature superconductor, La(2-)(x)Sr(x)CuO(4), where incommensurate peaks persist well above T(c). The temperature-dependent incommensurability is difficult to reconcile with interpretations based on charge stripe formation in YBa(2)Cu(3)O(6+x) near optimum doping.