The distribution of displacements in a fluid of hard disks is found by molecular dynamics to be non-Gaussian in the long-time limit, as surmised from the moments of the distribution that yield divergent diffusion and Burnett coefficients. On the other hand, for the Lorentz gas of hard disks, the distribution of displacements is Gaussian in the long-time limit and the diffusion coefficient exists, though the autocorrelation functions have power law tails, which lead to divergent Burnett coefficients.