Combination chemotherapy can cure certain leukemias and lymphomas, but most solid cancers are only curable at early stages. We review quantitative principles that explain the benefits of combining independently active cancer therapies in both settings. Understanding the mechanistic principles underlying curative treatments, including those developed many decades ago, is valuable for improving future combination therapies. We discuss contemporary evidence for long-established but currently neglected ideas of how combination therapy overcomes tumor heterogeneity. We show that a unified model of interpatient and intratumor heterogeneity describes historical progress in the treatment of pediatric acute lymphocytic leukemia (ALL), in which increasingly intensive combination regimens ultimately achieved high cure rates. We also describe three distinct aspects of drug independence that apply at different biological scales. The ability of these principles to quantitatively explain curative regimens suggests that supra-additive (synergistic) drug interactions are not required for successful combination therapy.