The impact of the now widely acknowledged self-similar property of network traffic on cell delay in a single server queuing model is investigated. The analytic traffic model, called N-Burst, uses the superposition of N independent cell streams of ON/OFF type with Power-Tail distributed ON periods. Delay for such arrival processes is mainly caused by over-saturation periods, which occur when too many sources are in their ON-state. The duration of these over- saturation periods is shown to have a Power-Tail distribution, whose exponent (beta) is in most scenarios different from the tail exponent of the individual ON- period. Conditions on the model parameters, for which the mean and higher moments of the delay distribution become infinite, are investigated. Since these conditions depend on traffic parameters as well as on network parameters, careful network design can alleviate the performance impact of such self-similar traffic. Furthermore, in real networks, a Maximum Burst Size (MBS) leads to truncated tails. An asymptotic relationship between the delay moments and the MBS is derived and is validated by the exact numerical results of the analytic queuing model.