Band convergence is considered a clear benefit to thermoelectric performance because it increases the charge carrier concentration for a given Fermi level, which typically enhances charge conductivity while preserving the Seebeck coefficient. However, this advantage hinges on the assumption that interband scattering of carriers is weak or insignificant. With first-principles treatment of electron-phonon scattering in the CaMg2Sb2-CaZn2Sb2 Zintl system and full Heusler Sr2SbAu, we demonstrate that the benefit of band convergence can be intrinsically negated by interband scattering depending on the manner in which bands converge. In the Zintl alloy, band convergence does not improve weighted mobility or the density-of-states effective mass. We trace the underlying reason to the fact that the bands converge at a one k-point, which induces strong interband scattering of both the deformation-potential and the polar-optical kinds. The case contrasts with band convergence at distant k-points (as in the full Heusler), which better preserves the single-band scattering behavior thereby successfully leading to improved performance. Therefore, we suggest that band convergence as thermoelectric design principle is best suited to cases in which it occurs at distant k-points.