Studies seeking to identify a human-caused global warming signal generally rely on climate model estimates of the “noise” of intrinsic natural variability. Assessing the reliability of these noise estimates is of critical importance. We evaluate here the statistical significance of differences between climate model and observational natural variability spectra for global-mean mid- to upper-tropospheric temperature (TMT). We use TMT information from satellites and large multimodel ensembles of forced and unforced simulations. Our main goal is to explore the sensitivity of model-versus-data spectral comparisons to a wide range of subjective decisions. These include the choice of satellite and climate model TMT datasets, the method for separating signal and noise, the frequency range considered, and the statistical model used to represent observed natural variability. Of particular interest is the amplitude of the interdecadal noise against which an anthropogenic tropospheric warming signal must be detected. We find that on time scales of 5–20 years, observed TMT variability is (on average) overestimated by the last two generations of climate models participating in the Coupled Model Intercomparison Project. This result is relatively insensitive to different plausible analyst choices, enhancing confidence in previous claims of detectable anthropogenic warming of the troposphere and indicating that these claims may be conservative. A further key finding is that two commonly used statistical models of short-term and long-term memory have deficiencies in their ability to capture the complex shape of observed TMT spectra.