For an upcoming paper, lots of cubic graphs up to 30 nodes have been examined. Crossing Number Web Compute and QuickCross are two programs that can calculate crossing numbers.
For example, the rectilinear crossing number of the 16-cell graph is 8, but the crossing number is 6.
So far, with cubic graphs, up to 30 nodes, any time a graph has been verified to have a particular crossing number, a rectilinear embedding with that crossing number has been found. Is that always the case?