We show that the band gap bowing trends observed in III-V alloys containing dilute concentrations of Sb or Bi can be explained within the framework of the valence-band anticrossing model. Hybridization of the extended p-like states comprising the valence band of the host semiconductor with the close-lying localized p-like states of Sb or Bi leads to a nonlinear shift of the valence-band edge and a reduction of the band gap. The two alloys GaSbxAs1−x and GaBixAs1−x are explored in detail, and the results are extrapolated to additional systems.