In this note, we first study a class of nonlinear interconnected systems and derive sufficient conditions for their global asymptotic stability in the presence of time delays. These conditions give estimates for the maximum admissible delays that the nominal delay-free model can tolerate without losing stability. 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 Kelly's network flow model of the Internet, and code-division multiple access (CDMA) uplink power control of cellular networks.