The Gröbner fan of an ideal $I\subset k[x_1,\dots,x_n]$, defined by Mora and Robbiano, is a complex of polyhedral cones in $\R^n$. The maximal cones of the fan are in bijection with the distinct monomial initial ideals of $I$ as the term order varies. If $I$ is homogeneous the Gröbner fan is complete and is the normal fan of the state polytope of $I$. In general the Gröbner fan is not complete and therefore not the normal fan of a polytope. We may ask if the restricted Gröbner fan, a subdivision of $\mathbb{R}_{\geq 0}^n$, is regular i.e. the normal fan of a polyhedron. The main result of this paper is an example of an ideal in $\mathbb{Q}[x_1,\dots,x_4]$ whose restricted Gröbner fan is not regular.