Well, be sure you take in the depth of the faces. Some may be too thin to cut, and can be cut into the actual mechanism.
I've made a representation of a Bump cube, and what it CAN look like if made into a fisher puzzle. (Not actual dimensions, but for the example right now)
bump.png [ 175.22 KiB | Viewed 885 times ]
So when rotated on one axis, it can look like this, if the same cuts are made.
bump2.png [ 197.59 KiB | Viewed 885 times ]
The problems appear when pieces like the one in Red, are really close to the exterior of the puzzle, and there is little holding it in already.
bump3.png [ 199.57 KiB | Viewed 885 times ]
The last part, is when the cuts are kept, and the piece has almost nothing left to grip onto (depending on what axis was rotated for the cuts). And the second problem are thin faces circled in purple, where there is very sharp edges (again, axis it was rotated upon matters).
bump4.png [ 193.18 KiB | Viewed 885 times ]
I'm not discouraging you from doing this, but I want to express that it can be dangerous. 3D printing is an option, so is extensions via cut and paste method, or sculpting and sanding. I'd love to see this be made, but make sure you know what angles you are cutting at, as you may have parts broken.