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In this paper we first study a class of nonlinear interconnected systems and derive sufficient conditions for their delay robustness. We then study a subclass with a specific interconnection structure that appears, among other applications, in several communication networks. To characterize their robustness to delays in backward and forward channels, we combine our main result with a passivity-based stability analysis, applicable to this subclass. We conclude with application examples from TCP/IP protocols in the Internet, and CDMA uplink power control in cellular networks.