Crystal nucleation is a much-studied phenomenon, yet the rate at which it occurs remains difficult to predict. Small crystal nuclei form spontaneously in supersaturated solutions, but unless their size exceeds a critical value--the so-called critical nucleus--they will re-dissolve rather than grow. It is this rate-limiting step that has proved difficult to probe experimentally. The crystal nucleation rate depends on Pcrit, the (very small) probability that a critical nucleus forms spontaneously, and on a kinetic factor (kappa) that measures the rate at which critical nuclei subsequently grow. Given the absence of a priori knowledge of either quantity, classical nucleation theory is commonly used to analyse crystal nucleation experiments, with the unconstrained parameters adjusted to fit the observations. This approach yields no 'first principles' prediction of absolute nucleation rates. Here we approach the problem from a different angle, simulating the nucleation process in a suspension of hard colloidal spheres, to obtain quantitative numerical predictions of the crystal nucleation rate. We find large discrepancies between the computed nucleation rates and those deduced from experiments: the best experimental estimates of Pcrit seem to be too large by several orders of magnitude.