A large number of products implementing aggregate buffering and scheduling mechanisms have been developed and deployed, and still more are under development. With the rapid increase in the demand for reliable end-to-end QoS solutions, it becomes increasingly important to understand the implications of aggregate scheduling on the resulting QoS capabilities. This document studies the bounds on the worst case delay in a network implementing aggregate scheduling. A lower bound on the worst case delay is derived. It is shown that in a general network configuration the delays achieved with aggregate scheduling are very sensitive to utilization and can in general be quite high. It is also shown that for a class of network configurations described in the paper it is possible to give an upper bound that provides reasonable worst case delay guarantees for reasonable utilization numbers. These bounds are a function of network utilization, maximum hop count of any flow, and the shaping parameters at the network ingress. It is also argued that for a general network configuration and utilization numbers which are independent on the maximum hop count, an upper bound on delay, if it exists, must be a function of the number of nodes and/or the number of flows in the network.