The cosmic microwave background (CMB) places a variety of model-independent constraints on the strength interactions of the dominant component of dark matter with the standard model. Percent-level subcomponents of the dark matter can evade the most stringent CMB bounds by mimicking the behavior of baryons, allowing for larger couplings and novel experimental signatures. However, in this Letter, we will show that such tightly coupled subcomponents leave a measurable imprint on the CMB that is well approximated by a change to the helium fraction, Y_{He}. Using the existing CMB constraint on Y_{He}, we derive a new upper limit on the fraction of tightly coupled dark matter, f_{TCDM}, of f_{TCDM}<0.006 (95% C.I.). We show that future CMB experiments can reach f_{TCDM}<0.001 (95% C.I.) and confirm that the bounds derived in this way agree with the results of a complete analysis. These bounds provide an example of how CMB constraints on Y_{He} have applications beyond studying big bang nucleosynthesis, since tightly coupled dark matter plays no direct role in the formation of light nuclei. We briefly comment on the implications for model building, including millicharged dark matter.