The infinite sites assumption, which states that every genomic position mutates at most once over the lifetime of a tumor, is central to current approaches for reconstructing mutation histories of tumors, but has never been tested explicitly. We developed a rigorous statistical framework to test the assumption with single-cell sequencing data. The framework accounts for the high noise and contamination present in such data. We found strong evidence for recurrent mutations at the same site in 8 out of 9 single-cell sequencing datasets from human tumors. Six cases involved the loss of earlier mutations, five of which occurred at sites unaffected by large scale genomic deletions. Two cases exhibited parallel mutation, including the dataset with the strongest evidence of recurrence. Our results refute the general validity of the infinite sites assumption and indicate that more complex models are needed to adequately quantify intra-tumor heterogeneity.