If I am doing the computation properly, then one example arises from the tangent cone at the origin of the closed subset $C\subset \mathbb{A}^4$ with defining ideal $$I = \langle x,y\rangle \cap \langle z,w^2-x\rangle = \langle xz,yz,xw^2-x^2,yw^2-xy \rangle,$$ where $(x,y,z,w)$ are coordinates on $\mathbb{A}^4$. The tangent cone should have defining ideal, $J=\langle xz,yz,x^2,xy \rangle$, which is the equation of the union $Z(x,y)\cup Z(x,z)$, but with embedded structure on the intersection, $Z(x,y,z)$.