Combination of anti-cancer drugs is broadly seen as way to overcome the often-limited efficacy of single agents. The design and testing of combinations are however very challenging. Here we present a uniquely large dataset screening over 5000 targeted agent combinations across 81 non-small cell lung cancer cell lines. Our analysis reveals a profound heterogeneity of response across the tumor models. Notably, combinations very rarely result in a strong gain in efficacy over the range of response observable with single agents. Importantly, gain of activity over single agents is more often seen when co-targeting functionally proximal genes, offering a strategy for designing more efficient combinations. Because combinatorial effect is strongly context specific, tumor specificity should be achievable. The resource provided, together with an additional validation screen sheds light on major challenges and opportunities in building efficacious combinations against cancer and provides an opportunity for training computational models for synergy prediction.