The nonisotropic nature of the neutrino emission from a supernova (SN) core might potentially affect the flavor evolution of the neutrino ensemble, via neutrino-neutrino interactions in the deepest SN regions. We investigate the dependence of these ``multiangle effects{''} on the original SN neutrino fluxes in a threeflavor framework. We show that the pattern of the spectral crossings (energies where Fv(e) Fv(x), and F (v) over bar (e) F (v) over bar (x)) is crucial in determining the impact of multiangle effects on the flavor evolution. For neutrino spectra presenting only a single crossing, synchronization of different angular modes prevails over multiangle effects, producing the known ``quasi-single-angle{''} evolution. Conversely, in the presence of spectra with multiple crossing energies, synchronization is not stable. In this situation, multiangle effects would produce a sizable delay in the onset of the flavor conversions, as recently observed. We show that, due to the only partial adiabaticity of the evolution at large radii, the multiangle suppression can be so strong to dramatically affect the final oscillated neutrino spectra. In particular three-flavor effects, associated with the solar parameters, could be washed out in multiangle simulations.