Drug combinations have been proposed to combat drug resistance in cancer, but due to the large number of possible drug targets, testing of all possible combinations of drugs is challenging. Computational models of a disease hold great promise as tools for prediction of response to treatment, and here we constructed a logical model integrating signaling pathways frequently dysregulated in cancer, as well as pathways activated upon DNA damage, to study the effect of clinically relevant drug combinations. By fitting the model to a dataset of pairwise combinations of drugs targeting MEK, PI3K, and TAK1, as well as several clinically approved agents (palbociclib, olaparib, oxaliplatin, and 5FU), we were able to perform model simulations that allowed us to predict more complex drug combinations, encompassing sets of three and four drugs, with potentially stronger effects compared to pairwise drug combinations. All predicted third-order synergies, as well as a subset of non-synergies, were successfully confirmed by experiments in the colorectal cancer cell line HCT-116, highlighting the strength of using computational strategies to rationalize drug testing.