Despite being widely and successfully applied to study transport in biophysical systems, the statistics of single-particle tracking data are only partially understood even in the simplest case of free diffusion. Here, we present the correct distribution of measurement results for a freely diffusing particle observed with localization error sigma and a finite camera integration time. We derive the fundamental limit (Cramer-Rao bound) on the error in estimating the diffusion coefficient D from such data, represented by a simple formula that can be applied to judge whether experimental data contains enough information to determine D. Two recently developed estimation procedures, a maximum-likelihood estimator [A. J. Berglund, Phys. Rev. E 82, 011917 (2010)] and an optimized least-squares fit to the mean-square displacement [X. Michalet, Phys. Rev. E 82, 041914 (2010)], are shown through numerical simulations to be nearly optimal in extracting D and sigma. These results can be applied to understand when D can be determined with reasonable confidence from short trajectories or in high-noise scenarios.