We report the likely identification of a substantial population of massive M ~ 10sup 11 M galaxies at z ~ 4 with suppressed star formation rates (SFRs), selected on rest-frame optical to near-IR colors from the FourStar Galaxy Evolution Survey (ZFOURGE). The observed spectral energy distributions show pronounced breaks, sampled by a set of near-IR medium-bandwidth filters, resulting in tightly constrained photometric redshifts. Fitting stellar population models suggests large Balmer/4000A breaks, relatively old stellar populations, large stellar masses, and low SFRs, with a median specific SFR of 2.9 + or -1.8 10sup -11 yrsup -1. Ultradeep Herschel/PACS 100 mu m, 160 mu m and Spitzer/MIPS 24 mu m data reveal no dust-obscured SFR activity for 15/19(79%) galaxies. The study suggests that most of the star formation in the progenitors of quiescent z ~ 4 galaxies was obscured by dust.