Abstract Recently, the first JWST measurement of thermal emission from a rocky exoplanet was reported. The inferred dayside brightness temperature of TRAPPIST-1 b at 15 μ m is consistent with the planet having no atmosphere and therefore no mechanism by which to circulate heat to its nightside. In this Letter, we compare TRAPPIST-1 b's measured secondary eclipse depth to predictions from a suite of self-consistent radiative-convective equilibrium models in order to quantify the maximum atmospheric thickness consistent with the observation. We find that plausible atmospheres (i.e., those that contain at least 100 ppm CO 2 ) with surface pressures greater than 0.3 bar are ruled out at 3 σ , regardless of the choice of background atmosphere, and a Mars-like thin atmosphere with surface pressure 6.5 mbar composed entirely of CO 2 is also ruled out at 3 σ . Thicker atmospheres of up to 10 bar (100 bar) are consistent with the data at 1 σ (3 σ ) only if the atmosphere lacks any strong absorbers across the mid-IR wavelength range—a scenario that we deem unlikely. We additionally model the emission spectra for bare-rock planets of various compositions. We find that a basaltic, metal-rich, and Fe-oxidized surface best matches the measured eclipse depth to within 1 σ , and the best-fit gray albedo is 0.02 ± 0.11. We conclude that planned secondary eclipse observations at 12.8 μ m will serve to validate TRAPPIST-1 b's high observed brightness temperature, but are unlikely to further distinguish among the consistent atmospheric and bare-rock scenarios.