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We present results from the MOSFIRE Deep Evolution Field survey on broad flux from the nebular emission lines Hα, [N ii], [O iii], Hβ, and [S ii]. The sample consists of 127 star-forming galaxies at 1.37 < z < 2.61 and 84 galaxies at 2.95 < z < 3.80. We decompose the emission lines using narrow and broad Gaussian components that we define as having FWHM < 275 km s−1 and FWHM > 300 km s−1, respectively, for both individual galaxies and stacks. For individual galaxies, broad emission is detected at >3σ in <10% of galaxies and the broad flux accounts for 10%–70% of the total flux. In the stacks, we find a slight increase in broad to narrow flux ratio with mass but note that we cannot reliably detect broad emission with FWHM < 275 km s−1, which may be significant at low masses. When placed on the N2-BPT diagram ([O iii]/Hβ versus [N ii]/Hα), the broad components of the stacks are shifted toward higher [O iii]/Hβ and [N ii]/Hα ratios compared to the narrow component. We compare the location of the broad components to shock models and find that the broad component could be explained as a shocked outflow, but we do not rule out other possibilities, such as the presence of an AGN. We discuss the possible consequences of shocked emission on the galaxy location in emission line diagnostic diagrams and calculation of SFR. We attempt to estimate the mass outflow rate/star formation rate, but caution that our results strongly rely on the assumptions regarding the physical properties of the outflow. [ABSTRACT FROM AUTHOR]