Let $\pi$ be an algebraic automorphic representation of a reductive group $G$ over a totally real number field $F$. Assume $G$ is anisotropic at infinity, and $\pi$ is not congruent to an automorphic character. Suppose $w$ is a finite place of $F$ where the component of $\pi$ is unramified and congruent to the trivial representation. Then there is an automorphic representation $\pi'$ of $G$ congruent to $\pi$, with the same central character and type at infinity, whose component at w is more ramified than that of $\pi$. Applications in rank one and two include showing that Saito-Kurokawa forms are congruent to generic ones, for the genus two symplectic group.