Bilevel optimization studies problems where the optimal response to a second mathematical optimization problem is integrated in the constraints. Such structure arises in a variety of decision-making problems in areas such as market equilibria, policy design or product pricing. We introduce near-optimal robustness for bilevel problems, protecting the upper-level decision-maker from bounded rationality at the lower level and show it is a restriction of the corresponding pessimistic bilevel problem. Essential properties are derived in generic and specific settings. This model finds a corresponding and intuitive interpretation in various situations cast as bilevel optimization problems. We develop a duality-based solution method for cases where the lower level is convex, leveraging the methodology from robust and bilevel literature. The models obtained are tested numerically using different solvers and formulations, showing the successful implementation of the near-optimal bilevel problem.