Experiments show that the transit times of chirped, narrow-band pulses that move across nonabsorbing, one-dimensional barriers are modified dramatically by the interplay between the chirp and the transmission function of the sample. In an experiment we monitored 0.9-ns chirped, nearly Gaussian pulses as they traversed a 450-mum GaAs etalon. At certain wavelengths pulse transit times can be superluminal or even negative. To explain these phenomena we have proposed a generalization of the transit time for chirped pulses that is still meaningful even when the transit times are superluminal or negative. Our predictions agree well with the experimental results.