Many theories for the Madden-Julian oscillation (MJO) focus on diabatic processes, particularly the evolution of vertical heating and moistening. Poor MJO performance in weather and climate models is often blamed on biases in these processes and their interactions with the large-scale circulation. We introduce one of the three components of a model evaluation project, which aims to connect MJO fidelity in models to their representations of several physical processes, focusing on diabatic heating and moistening. This component consists of 20day hindcasts, initialized daily during two MJO events in winter 2009-2010. The 13 models exhibit a range of skill: several have accurate forecasts to 20days lead, while others perform similarly to statistical models (8-11days). Models that maintain the observed MJO amplitude accurately predict propagation, but not vice versa. We find no link between hindcast fidelity and the precipitation-moisture relationship, in contrast to other recent studies. There is also no relationship between models' performance and the evolution of their diabatic heating profiles with rain rate. A more robust association emerges between models' fidelity and net moistening: the highest-skill models show a clear transition from low-level moistening for light rainfall to midlevel moistening at moderate rainfall and upper level moistening for heavy rainfall. The midlevel moistening, arising from both dynamics and physics, may be most important. Accurately representing many processes may be necessary but not sufficient for capturing the MJO, which suggests that models fail to predict the MJO for a broad range of reasons and limits the possibility of finding a panacea.