This paper examines the effects of boundedly rational decision characteristics on travelers’ route choice behavior. The concept of boundedly rational confidence level (BRCL) is redefined, which is the probability that a trip arrives between the acceptable earliest arrival time and the acceptable latest arrival time on the shortest travel time budget (TTB). Mathematically, the acceptable boundedly rational arrival thresholds are proposed. Then, a reliability-based boundedly rational traffic equilibrium model (R-BRTE) considering both travel time reliability and acceptable arrival thresholds is developed. Moreover, the equivalent variational inequality problem and uniqueness of solution on the proposed model are proved. A route-based solution algorithm is used to solve the proposed R-BRTE model. Numerical results present the important decision ideas of the proposed model. The results demonstrate that travelers’ bounded rationality has a great impact on their route choice behavior and network performance.