Most natural task-relevant variables are encoded in the early sensory cortex in a form that can only be decoded nonlinearly. Yet despite being a core function of the brain, nonlinear population codes are rarely studied and poorly understood. Interestingly, the most relevant existing quantitative model of nonlinear codes is inconsistent with known architectural features of the brain. In particular, for large population sizes, such a code would contain more information than its sensory inputs, in violation of the data processing inequality. In this model, the noise correlation structures provide the population with an information content that scales with the size of the cortical population. This correlation structure could not arise in cortical populations that are much larger than their sensory input populations. Here we provide a better theory of nonlinear population codes that obeys the data processing inequality by generalizing recent work on information-limiting correlations in linear population codes. Although these generalized, nonlinear information-limiting correlations bound the performance of any decoder, they also make decoding more robust to suboptimal computation, allowing many suboptimal decoders to achieve nearly the same efficiency as an optimal decoder. Although these correlations are extremely difficult to measure directly, particularly for nonlinear codes, we provide a simple, practical test by which one can use choice-related activity in small populations of neurons to determine whether decoding is limited by correlated noise or by downstream suboptimality. Finally, we discuss simple sensory tasks likely to require approximately quadratic decoding, to which our theory applies.