Consider the 3-simplex, or tetrahedron, in 3-space. Regardless of the positions of the vertices, every point in the volume defined by the triangular faces of the polytope's skeleton graph can lie along a chord between two non-adjacent edges. Or, equivalently, every interior point can lie along a straight line segment which intersects two non-adjacent edges.
When is this property true of other convex (or non-convex) polyhedra? How does this property extend to the general $N$-simplex?
Addendum [10/31/2011]: Considering Igor Pak's very nice answer, I'm wondering if it's possible to derive some hard lower bounds on the number of distinct chords passing through an arbitrary point $v$ interior to a given polytope. I could certainly be mistaken, but it seems to me that there always must be at least two solutions.