The consistency of tropical tropospheric temperature trends with climate model expectations remains contentious. A key limitation is that the uncertainties in observations from radiosondes are both substantial and poorly constrained. We present a thorough uncertainty analysis of radiosonde‐based temperature records. This uses an automated homogenization procedure and a previously developed set of complex error models where the answer is known a priori. We perform a number of homogenization experiments in which error models are used to provide uncertainty estimates of real‐world trends. These estimates are relatively insensitive to a variety of processing choices. Over 1979–2003, the satellite‐equivalent tropical lower tropospheric temperature trend has likely (5–95% confidence range) been between −0.01 K/decade and 0.19 K/decade (0.05–0.23 K/decade over 1958–2003) with a best estimate of 0.08 K/decade (0.14 K/decade). This range includes both available satellite data sets and estimates from models (based upon scaling their tropical amplification behavior by observed surface trends). On an individual pressure level basis, agreement between models, theory, and observations within the troposphere is uncertain over 1979 to 2003 and nonexistent above 300 hPa. Analysis of 1958–2003, however, shows consistent model‐data agreement in tropical lapse rate trends at all levels up to the tropical tropopause, so the disagreement in the more recent period is not necessarily evidence of a general problem in simulating long‐term global warming. Other possible reasons for the discrepancy since 1979 are: observational errors beyond those accounted for here, end‐point effects, inadequate decadal variability in model lapse rates, or neglected climate forcings.