Planning, scheduling, and other applications of heuristic search often demand we tackle problems that are too large to solve optimally. In this paper, we address the problem of solving shortest-path problems as quickly as possible while guaranteeing that solution costs are bounded within a specified factor of optimal. 38 years after its publication, weighted A* remains the best-performing algorithm for general-purpose bounded suboptimal search. However, it typically returns solutions that are better than a given bound requires. We show how to take advantage of this behavior to speed up search while retaining bounded suboptimality. We present an optimistic algorithm that uses a weight higher than the user's bound and then attempts to prove that the resulting solution adheres to the bound. While simple, we demonstrate that this algorithm consistently surpasses weighted A* in four different benchmark domains including temporal planning and gridworld pathfinding.