Australian rainfall is strongly influenced by El Nino-southern oscillation (ENSO). The relationship between ENSO and rainfall in eastern Australia is non-linear; the magnitude of La Nina events has a greater effect on rainfall than does the magnitude of El Nino events, and the cause of the non-linearity is unclear from previous work. The twentieth century reanalysis succeeds in capturing the asymmetric ENSO-rainfall relationship. In the reanalysis the asymmetry is strongly related to moisture availability in the south-west Pacific whereas wind flow is of less importance. Some global climate models (GCMs) in the coupled model intercomparison project (CMIP5) archive capture the asymmetric nature of the ENSO-rainfall relationship whilst others do not. Differences in thermodynamic processes and their relationships with ENSO are the primary cause of variability in model performance. Analysis of an atmosphere-only run of a GCM which fails to capture the non-linear ENSO-rainfall relationship is also conducted. The atmospheric run forced by observed sea surface temperatures shows no significant improvement in the ENSO-rainfall relationship over the corresponding coupled model run in the CMIP5 archive. This result suggests that some models are failing to capture the atmospheric teleconnection between the tropical Pacific and Australia, and both this and a realistic representation of oceanic ENSO characteristics are required for coupled models to accurately capture the ENSO-rainfall teleconnection. These findings have implications for the study of rainfall projections in the region.