A fundamental aspect of the observed ENSO is the positive asymmetry between its two phases: the strongest El Niño is stronger than the strongest La Niña. The nonlinear term in the equation for the surface ocean heat budget has been theorized as a cause of the asymmetry. This theory is challenged by the diversity of asymmetry among the CMIP5 models: these models all employ primitive equations and thus have the nonlinear term in the heat budget equation for the ocean surface, yet the asymmetry simulated by these models ranges from significantly negative to significantly positive. Here, the authors employ an analytical but nonlinear model—a model that simulates the observed ENSO asymmetry—to show that the nonlinear heating term does not guarantee the oscillation in the system to possess positive asymmetry. Rather, the system can have regimes with negative, zero, and positive asymmetry. The regime in which the system finds itself depends on a multitude of physical parameters. Moreover, the range of certain physical parameters for the system to fall in the regime with positive asymmetry in the oscillation is rather narrow, underscoring the difficulty of simulating the observed ENSO asymmetry by CMIP5 models. Moreover, stronger positive asymmetry is found to be associated with a more complicated oscillation pattern: the two adjacent strongest warm events are spaced farther apart and more small events occur in between. These results deepen the understanding of factors that are behind the asymmetry of ENSO and offer paths to take to improve model-simulated ENSO asymmetry.