Model diagnostic procedures in failure time models using hazard-based residuals rely on the assumption that the residual vector closely resembles a random sample from a unit exponential distribution when the model holds. This article formally investigates the validity of this critical assumption by deriving and examining the properties of parametrically, semiparametrically, and nonparametrically estimated residuals for complete and right-censored data. The joint distribution of the residual vector is characterized, and the behavior of some tests for exponentiality when applied to the residuals is examined analytically and through Monte Carlo methods. Findings reveal that the critical assumption of approximate unit exponentiality of the residual vector may not be viable and, consequently, the model diagnostic procedures considered, which revolve on checking the approximate unit exponentiality of the residual vector (specifically, hazard plotting and the use of spacings and total-time-on-test statistics on the residual vector) may have serious defects. This is especially evident in situations where the failure time distribution is not exponential or when the residuals are obtained nonparametrically in the no-covariate model or semiparametrically in the Cox proportional hazards model.