Gene expression involves inherently probabilistic steps that create fluctuations in protein abundances. The results from many in-depth analyses and genome-scale surveys have suggested how such fluctuations arise and spread, often in ways consistent with stochastic models of transcription and translation. But fluctuations also arise during cell division when molecules are partitioned stochastically between the two daughters. Here we mathematically demonstrate how stochastic partitioning contributes to the non-genetic heterogeneity. Our results show that partitioning errors are hard to correct, and that the resulting noise profiles are remarkably difficult to separate from gene expression noise. By applying these results to common experimental strategies and distinguishing between creation versus transmission of noise, we hypothesize that much of the cell-to-cell heterogeneity that has been attributed to various aspects of gene expression instead comes from random segregation at cell division. We propose experiments to separate between these two types of fluctuations and discuss future directions.