We consider a popular approach to multicategory classification tasks: a two-stage system based on a first classifier with rejection followed by a nearest-neighbor classifier. Patterns which are not rejected by the first classifier are classified according to its output. Rejected patterns are passed to the nearest-neighbor classifier together with the top-h ranking classes returned by the first classifier. The nearest-neighbor classifier, looking at patterns in the top-h classes, classifies the rejected pattern. An editing strategy for the nearest-neighbor reference database, controlled by the first classifier, is also considered. We analyze this system. Moreover, we formally relate the response time of the system to the rejection rate of the first classifier and to the other system parameters. The error-response time trade-off is also discussed. Finally, we experimentally study two instances of the system applied to the recognition of handwritten digits. In one system, the first classifier is a fuzzy basis functions network, while in the second system it is a feed-forward neural network. Classification results as well as response times for different settings of the system parameters are reported for both systems.