Complex networks often have a modular structure, where a number of tightly- connected groups of nodes (modules) have relatively few interconnections. Modularity had been shown to have an important effect on the evolution and stability of biological networks, on the scalability and efficiency of large-scale infrastructure, and the development of economic and social systems. An analytical framework for understanding modularity and its effects on network vulnerability is still missing. Through recent advances in the understanding of multilayer networks, however, it is now possible to develop a theoretical framework to systematically study this critical issue. Here we study, analytically and numerically, the resilience of modular networks under attacks on interconnected nodes, which exhibit high betweenness values and are often more exposed to failure. Our model provides new understandings into the feedback between structure and function in real world systems, and consequently has important implications as diverse as developing efficient immunization strategies, designing robust large-scale infrastructure, and understanding brain function.