The protein homeostasis (proteostasis) network is composed of multiple pathways that work together to balance protein folding, stability, and turnover. Cancer cells are particularly reliant on this network; however, it is hypothesized that inhibition of one node might lead to compensation. To better understand these connections, we dosed 22Rv1 prostate cancer cells with inhibitors of four proteostasis targets (Hsp70, Hsp90, proteasome, and p97), either alone or in binary combinations, and measured the effects on cell growth. The results reveal a series of additive, synergistic, and antagonistic relationships, including strong synergy between inhibitors of p97 and the proteasome and striking antagonism between inhibitors of Hsp90 and the proteasome. Based on RNA-seq, these relationships are associated, in part, with activation of stress pathways. Together, these results suggest that cocktails of proteostasis inhibitors might be a powerful way of treating some cancers, although antagonism that blunts the efficacy of both molecules is also possible.