A three-dimensional LUT uses less input values in each channel than a one-dimensional LUT because the total size of the table raises with the cube of the number of input values. Systems that apply a 3DLUT interpolate between the values in the table.
As a consequence of this interpolation, an image may change slightly if the number of mesh points in the 3DLUT is low, say 163 or 173.
On the right side you see a video image (Rec 709) processed by the ALEXA camera. Compare this image with several versions of the same image on the left side that have been generated by converting a Log C image with a 3DLUT in different sizes. You will not be able to see any difference between the original and the results of lookup tables with 653 or 333 mesh points. The image generated by the smallest LUT, however, differs slightly from the original.