Dead-time compensators (DTCs) are a family of classical controllers derived from the Smith Predictor (SP). Their main characteristic is that they explicitly employ the model of the open-loop process to feedback a predicted value of the non-delayed system, thus obtaining \textit{compensation} of the delay. Such a perfect compensation is not achievable in the case of time-varying delays. In this paper, we address stability analysis of a DTC structure in this situation, in addition to considering saturating actuators and disturbances of limited energy. Specific challenges related to the DTC closed loop are taken into account in the developed theoretical conditions, which are expressed in terms of linear matrix inequalities (LMIs) by using an adequate Lyapunov-Krasovskii functional (LKF) and generalized sector conditions. Furthermore, a new approach for the definition of the set of initial conditions in an augmented space in conjunction with the LKF is presented. Besides theoretical innovations, practical discussion about the relation between the tuning of DTC controllers and robustness for this class of systems is presented through numerical examples. An experimental application on a neonatal incubator prototype is carried out to emphasize the effectiveness of the results.