Compressed digital video is one of the most important traffic types in future integrated services networks. However, a network service that supports delay-sensitive video imposes many problems since compressed video sources are variable bit rate (VBR) with a high degree of burstiness. In this paper, we consider a network service that can provide deterministic guarantees on the minimum throughput and the maximum delay of VBR video traffic. A common belief is that due to the burstiness of VBR traffic, such a service will not be efficient and will necessarily result in low network utilization. We investigate the fundamental Limits and trade-offs in providing deterministic performance guarantees to video and use a set of 10 to 30 min. long MPEG-compressed video traces for evaluation. Contrary to conventional wisdom, we are able to show that, in many cases, a deterministic service can be provided to video traffic while maintaining a reasonable level of network utilization. We first consider an ideal network environment that employs the most accurate deterministic, time-invariant video traffic characterizations, the optimal earliest-deadline-first packet schedulers, and exact admission control conditions. The utilization achievable in this situation provides the fundamental limits of a deterministic service. We then investigate the utilization limits in a network environment that takes into account practical constraints, such as the need for simple and efficient policing mechanisms, packet scheduling algorithms, and admission control tests.