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A networked dynamical system is composed of subsystems interconnected via prescribed interactions. In many engineering applications, however, one subsystem can also affect others via unintended interactions that can significantly change the intended network's behavior. Although unintended interactions can be modeled as disturbance inputs to the subsystems, these disturbances depend on the network's states. Thus, a disturbance attenuation property of each subsystem is insufficient to ensure that the network is robust to unintended interactions. Here, we provide conditions on subsystem dynamics and interaction maps, such that a network is robust to unintended interactions. These conditions require that each subsystem asymptotically attenuates constant external disturbances, is monotone or near-monotone, the unintended interactions are monotone, and the prescribed interactions do not contain feedback loops. We apply this result to guide the design of resource-limited genetic circuits composed of feedback-regulated subsystems.