A series of hexagons, three to a vertex, is flat, since 3x120 exactly = 360 (a full circle). There are many other sets of shapes that won't make polyhedra. Four squares to a corner, or six equilateral triangles to a corner won't for the same reason; the angles add up to 360 degrees. read more
If you relax your conditions so that the hexagons need not be regular and all vertices need not be the same you can consider the Euler characteristic V - E + F of the polyhedron. For the polyhedron not to be self-intersecting the characteristic must be 2. read more
(Hexagons are special. One way to interpret this result is that an infinite plane can be tiled with hexagons, so hexagons correspond to zero curvature, whereas since pentagons have a smaller angle at each vertex they correspond to positive curvature. What the above statement says, roughly, is that the total amount of curvature is a constant. read more