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BET inhibitors are promising therapeutic agents for the treatment of triple-negative breast cancer (TNBC), but the rapid emergence of resistance necessitates investigation of combination therapies and their effects on tumor evolution. Here, we show that palbociclib, a CDK4/6 inhibitor, and paclitaxel, a microtubule inhibitor, synergize with the BET inhibitor JQ1 in TNBC lines. High-complexity DNA barcoding and mathematical modeling indicate a high rate of de novo acquired resistance to these drugs relative to pre-existing resistance. We demonstrate that the combination of JQ1 and palbociclib induces cell division errors, which can increase the chance of developing aneuploidy. Characterizing acquired resistance to combination treatment at a single cell level shows heterogeneous mechanisms including activation of G1-S and senescence pathways. Our results establish a rationale for further investigation of combined BET and CDK4/6 inhibition in TNBC and suggest novel mechanisms of action for these drugs and new vulnerabilities in cells after emergence of resistance.