We investigate the hypothesis that Coulomb-type interactions between dark matter (DM) and baryons explain the anomalously low 21 cm brightness-temperature minimum at redshift z∼17 that was recently measured by the EDGES experiment. In particular, we reassess the validity of the scenario where a small fraction of the total DM is millicharged, focusing on newly derived constraints from Planck 2015 cosmic microwave background (CMB) data. Crucially, the CMB power spectrum is sensitive to DM-baryon scattering if the fraction of interacting DM is larger than (or comparable to) the fractional uncertainty in the baryon energy density. Meanwhile, there is a mass-dependent lower limit on the fraction for which the required interaction to cool the baryons sufficiently is so strong that it drives the interacting-DM temperature to the baryon temperature prior to their decoupling from the CMB. If this occurs as early as recombination, the cooling saturates. We precisely determine the viable parameter space for millicharged DM, and find that only a fraction (mχ/MeV)0.0115%f0.4% of the entire DM content, and only for DM-particle masses between 0.5 MeV-35 MeV, can be charged at the level needed to marginally explain the anomaly, without violating limits from SLAC, CMB, big bang nucleosynthesis (BBN), or stellar and SN1987A cooling. In reality, though, we demonstrate that at least moderate fine tuning is required to both agree with the measured absorption profile and overcome various astrophysical sources of heating. Finally, we point out that a ∼0.4% millicharged DM component which is tightly coupled to the baryons at recombination may resolve the current 2σ tension between the BBN and CMB determinations of the baryon energy density. Future CMB-S4 measurements will be able to probe this scenario directly.