Ed,

Not to speak for Mike's experience, but we used finely-set block planes to
thin the overlap spots on the boxes (i.e., put a bevel on the non-finger
ends of the bands for sides and tops).  We were getting it down to under
1/16", using cherry and maple, primarily, though I have a set of Ralph-made
cherry/black walnut boxes with the same sort of bevels.

Best,
Bill
Felton, CA