We consider a multi-priority, nonpreemptive, multi-server queueing system, with Poisson arrivals and negative exponential service times. In order to save available servers for higher priority customers, the system deliberately queues arriving lower priority customers whenever the number of busy servers equals or exceeds a given priority-dependent cutoff number. A queued, priority i customer enters service the instant there are fewer than the respective cutoff number of servers busy and all higher priority queues are empty. We derive several performance measures for the waiting time of the ith priority class: its mean, its second moment, and its distribution (in transforms). We also extend the analysis to systems in which any subset of priority levels may overflow to some other system rather than join infinite capacity queues.