The infinitesimal model of quantitative genetics relies on the Central Limit Theorem to stipulate that under additive models of quantitative traits determined by many loci having similar effect size, the difference between an offspring's genetic trait component and the average of their two parents' genetic trait components is Normally distributed and independent of the parents' values. Here, we investigate how the assumption of similar effect sizes affects the model: in particular, if the tail of the effect size distribution is polynomial with exponent α < 2, then sums of effects should be well-approximated by a "stable distribution", and find tail exponents between 1 and 2 in effect sizes estimated by genome-wide association studies of many human disease-related traits. We show that the independence of offspring trait deviations from parental averages in many cases implies a Gaussian distribution, suggesting that non-Gaussian models of trait evolution must explicitly track the underlying genetics, at least for loci of large effect. We also characterize possible limiting trait distributions of the infinitesimal model with infinitely divisible noise distributions, and compare our results to simulations.