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 μm 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 CO2) with surface pressures greater than 0.3 bar are ruled out at 3σ, regardless of the choice of background atmosphere, and a Mars-like thin atmosphere with surface pressure 6.5 mbar composed entirely of CO2 is also ruled out at 3σ. Thicker atmospheres of up to 10 bar (100 bar) are consistent with the data at 1σ (3σ) 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 1σ, and the best-fit gray albedo is 0.02 ± 0.11. We conclude that planned secondary eclipse observations at 12.8 μm 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.