In this paper, we provide a new nonconservative upper bound for the settling time of a class of fixed-time stable systems. To expose the value and the applicability of this result, we present four main contributions. First, we revisit the well-known class of fixed-time stable systems, to show the conservatism of the classical upper estimate of its settling time. Second, we provide the smallest constant that the uniformly upper bounds the settling time of any trajectory of the system under consideration. Third, introducing a slight modification of the previous class of fixed-time systems, we propose a new predefined-time convergent algorithm where the least upper bound of the settling time is set a priori as a parameter of the system. At last, we design a class of predefined-time controllers for first- and second-order systems based on the exposed stability analysis. Simulation results highlight the performance of the proposed scheme regarding settling time estimation compared to existing methods.