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Serafin J. Grundl and Yu Zhu | In laboratory experiments bidding in first-price auctions is more aggressive than predicted by the risk-neutral Bayesian Nash Equilibrium (RNBNE) - a finding known as the overbidding puzzle. Several models have been proposed to explain the overbidding puzzle, but no canonical alternative to RNBNE has emerged, and RNBNE remains the basis of the structural auction literature. Instead of estimating a particular model of overbidding, we use the overbidding restriction itself for identification, which allows us to bound the valuation distribution, the seller's payoff function, and the optimal reserve price. These bounds are consistent with RNBNE and all models of overbidding and remain valid if different bidders employ different bidding strategies. We propose simple estimators and evaluate the validity of the bounds numerically and in experimental data.