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We investigate scaling properties of end-to-end delays in packet networks for a flow that traverses a sequence of H nodes and that experiences cross traffic at each node. When the traffic flow and the cross traffic do not satisfy independence assumptions, we find that delay bounds scale faster than linearly. More precisely, for exponentially bounded packetized traffic, we show that delays grow with Θ(H logH) in the number of nodes on the network path. This superlinear scaling of delays is qualitatively different from the scaling behavior predicted by a worst-case analysis or by a probabilistic analysis assuming independence of traffic arrivals at network nodes.