We explore the effects of probabilistic RNA partitioning during cell division on the normalized variance of RNA numbers across generations of bacterial populations. We first characterize these effects in model cell populations, where gene expression is modeled as a delayed stochastic process, as a function of the synchrony in cell division, the rate of division, and the RNA degradation rate. We further explore the additional variance that arises if the partitioning is biased. Next, in Escherichia coli cells expressing RNA tagged with MS2d-GFP, we measured the normalized variance of RNA numbers across several generations, with cell divisions synchronized by heat shock. We show that synchronized cell populations exhibit transient increases in normalized variance following cell divisions, as predicted by the model, which are not observed in unsynchronized populations. We conclude that errors in partitioning of RNA molecules generate diversity between the offspring of individual bacteria and thus constitute a form of reproductive bet-hedging.