Cancer therapies often have narrow therapeutic indexes and involve potentially suboptimal combinations due to the dissimilar physical properties of drug molecules. Nanomedicine platforms could address these challenges, but it remains unclear whether synergistic free-drug ratios translate to nanocarriers and whether nanocarriers with multiple drugs outperform mixtures of single-drug nanocarriers at the same dose. Here we report a bottlebrush prodrug (BPD) platform designed to answer these questions in the context of multiple myeloma therapy. We show that proteasome inhibitor (bortezomib)-based BPD monotherapy slows tumour progression in vivo and that mixtures of bortezomib, pomalidomide and dexamethasone BPDs exhibit in vitro synergistic, additive or antagonistic patterns distinct from their corresponding free-drug counterparts. BPDs carrying a statistical mixture of three drugs in a synergistic ratio outperform the free-drug combination at the same ratio as well as a mixture of single-drug BPDs in the same ratio. Our results address unanswered questions in the field of nanomedicine, offering design principles for combination nanomedicines and strategies for improving current front-line monotherapies and combination therapies for multiple myeloma.