Aneuploidy is a hallmark of breast cancer; however, knowledge of how these complex genomic rearrangements evolve during tumorigenesis is limited. In this study, we developed a highly multiplexed single-nucleus sequencing method to investigate copy number evolution in patients with triple-negative breast cancer. We sequenced 1,000 single cells from tumors in 12 patients and identified 1–3 major clonal subpopulations in each tumor that shared a common evolutionary lineage. For each tumor, we also identified a minor subpopulation of non-clonal cells that were classified as metastable, pseudodiploid or chromazemic. Phylogenetic analysis and mathematical modeling suggest that these data are unlikely to be explained by the gradual accumulation of copy number events over time. In contrast, our data challenge the paradigm of gradual evolution, showing that the majority of copy number aberrations are acquired at the earliest stages of tumor evolution, in short punctuated bursts, followed by stable clonal expansions that form the tumor mass.