In the first part of this paper, we study RBSDEs in the case where the filtration is non-quasi-left-continuous and the lower obstacle is given by a predictable process. We prove the existence and uniqueness by using some results of optimal stopping theory in the predictable setting, some tools from general theory of processes as the Mertens decomposition of predictable strong supermartingale. In the second part, we introduce an optimal stopping problem indexed by predictable stopping times with the nonlinear predictable expectation induced by an appropriate backward stochastic differential equation (BSDE). We establish some useful properties of -supremartingales. Moreover, we show the existence of an optimal predictable stopping time, and we characterize the predictable value function in terms of the first component of RBSDEs studied in the first part.