The amount of information available in spectropolarimetric data is estimated. To this end, the intrinsic dimensionality of the data is inferred with the aid of a recently derived estimator based on nearest neighbor considerations and obtained applying the principle of maximum likelihood. We show in detail that the estimator correctly captures the intrinsic dimension of artificial data sets with known dimension. The effect of noise in the estimated dimension is analyzed thoroughly, and we conclude that it introduces a positive bias that needs to be accounted for. Real simultaneous spectropolarimetric observations in the visible 630 nm and the near-infrared 1.5 μm spectral regions are also investigated in detail, showing that the near-infrared data set provides more information of the physical conditions in the solar atmosphere than the visible data set. Finally, we demonstrate that the amount of information present in an observed data set is a monotonically increasing function of the number of available spectral lines.