It is easy to see that if one harmonic function vanishes on a given set in the Euclidean plane, then infinitely many linearly independent harmonic functions do. Perhaps surprisingly, this is no longer true in higher dimensions. We will show that exactly two linearly independent harmonic functions vanish on the zero set of a generically chosen harmonic homogeneous polynomial of degree two. This result holds on the level of germs (at the origin). We will also show that smooth hypersurfaces can exhibit the same phenomenon if one asks for harmonic functions defined on a fixed, large enough domain in Euclidean space.