It has recently been argued that the PAMELA, ATIC, and PPB-BETS data showing an anomalous excess of high-energy cosmic ray positrons and electrons might be explained by dark matter annihilating in the Galactic halo with a cross section resonantly enhanced compared to its value in the primeval plasma. We find that with a very large annihilation cross section the flash of energetic photons and electron-positron pairs expected from dark-matter annihilation in the first protohalos that form at redshift $z\ensuremath{\sim}40$ is likely substantial and observable. As a consequence, bounds on the allowed energy injection into the primordial gas and the energy density of the diffuse gamma-ray background give rise to limits on the low-velocity dark-matter cross section that can be difficult to reconcile with this interpretation of the PAMELA, ATIC, and PPB-BETS results.