Computational strategies used by the brain strongly depend on the amount of information that can be stored in population activity, which in turn strongly depends on the pattern of noise correlations. In vivo, noise correlations tend to be positive and proportional to the similarity in tuning properties. Such correlations are thought to limit information, which has led to the suggestion that decorrelation increases information. In contrast, we found, analytically and numerically, that decorrelation does not imply an increase in information. Instead, the only information-limiting correlations are what we refer to as differential correlations: correlations proportional to the product of the derivatives of the tuning curves. Unfortunately, differential correlations are likely to be very small and buried under correlations that do not limit information, making them particularly difficult to detect. We found, however, that the effect of differential correlations on information can be detected with relatively simple decoders.