High signal-to-noise, representative spectra of star-forming galaxies at z ~ 2, obtained via stacking, reveal a high-velocity component underneath the narrow Hα and [N II] emission lines. When modeled as a single Gaussian, this broad component has FWHM 1500 km s–1; when modeled as broad wings on the Hα and [N II] features, it has FWHM 500 km s–1. This feature is preferentially found in the more massive and more rapidly star-forming systems, which also tend to be older and larger galaxies. We interpret this emission as evidence of either powerful starburst-driven galactic winds or active supermassive black holes (SMBHs). If galactic winds are responsible for the broad emission, the observed luminosity and velocity of this gas imply mass outflow rates comparable to the star formation rate. On the other hand, if the broad-line regions of active black holes account for the broad feature, the corresponding black holes masses are estimated to be an order of magnitude lower than those predicted by local scaling relations, suggesting a delayed assembly of SMBHs with respect to their host bulges.