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Let G be a p-group, and let χ be an irreducible character of G. The codegree of χ is given by |G : ker (χ)|/χ(1). Du and Lewis have shown that a p-group with exactly three codegrees has nilpotence class at most 2. Here we investigate p-groups with exactly four codegrees. If, in addition to having exactly four codegrees, G has two irreducible character degrees, G has largest irreducible character degree p2, |G : G′| = p2, or G has coclass at most 3, then G has nilpotence class at most 4. In the case of coclass at most 3, the order of G is bounded by p7. With an additional hypothesis, we can extend this result to p-groups with four codegrees and coclass at most 6. In this case, the order of G is bounded by p10. [ABSTRACT FROM AUTHOR]