We study contests where several privately informed agents bid for a prize. All bidders bear a cost of bidding that is an increasing function of their bids, and, moreover, bids may be capped. We show that regardless of the number of bidders, if bidders have linear or concave cost functions, then setting a bid cap is not profitable for a designer who wishes to maximize the average bid. On the other hand, if agents have convex cost functions (i.e., an increasing marginal cost), then effectively capping the bids is profitable for a designer facing a sufficiently large number of bidders. Furthermore, bid caps are effective for any number of bidders if the cost functions' degree of the convexity is large enough.