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Let $G$ be a $p$-group and let $\chi$ be an irreducible character of $G$. Thecodegree of $\chi$ is given by $|G:\text{ker}(\chi)|/\chi(1)$. Du and Lewishave shown that a $p$-group with exactly three codegrees has nilpotence classat most 2. Here we investigate $p$-groups with exactly four codegrees. If, inaddition to having exactly four codegrees, $G$ has two irreducible characterdegrees, $G$ has largest irreducible character degree $p^2$, $|G:G'|=p^2$, or$G$ has coclass at most 3, then $G$ has nilpotence class at most 4. In the caseof coclass at most 3, the order of $G$ is bounded by $p^7$. With an additionalhypothesis we can extend this result to $p$-groups with four codegrees andcoclass at most 7. In this case the order of $G$ is bounded by $p^{11}$.