To effectively handle the overlapping path issue in the multinomial logit (MNL) model and perfectly rational issue in the expected utility theory (EUT) while capturing the time-varying probabilistic distribution characteristics of origin—destination (OD) demand, this study develops a reliability-based dynamic traffic assignment (R-DTA) model, that is, a cumulative prospect value (CPV)-based generalized nested logit (GNL) stochastic user equilibrium (SUE) model under stochastic time-varying conditions. Specifically, the proposed R-DTA model is established by replacing the utility value with the CPV as the path performance within the GNL model framework. An equivalent variational inequality model is provided for the proposed R-DTA model, which is solved by the method of successive averages (MSA). Moreover, the existence and equivalence of the solution to the equivalent model are also proved. The proposed R-DTA model is tested on two networks to demonstrate its performance. The corresponding results demonstrate that the model can jointly deal with the perfectly rational issues and the overlapping path issues; also, the model can effectively capture the time-varying probabilistic distribution characteristics of OD demand.