Consider 28 $19\times44\times29$ cuboids in a box.
{19, 44, 29} {7, 2, 2} pack in a {133, 88, 58} box.
{44, 29, 19} {3, 3, 3} pack in a {132, 87, 57} box.
By reorienting the cuboids, one can be removed and the others seem to fit snugly in the box.
{43, 49, 65} {8, 4, 2} pack in a {344, 196, 130} box.
{49, 65, 43} {7, 3, 3} pack in a {343, 195, 129} box.
Twisting the cuboids lets one of them seem to vanish.
Is there a nice vanishing cuboid solution that uses all three cuboid orientations?
First orientation the $n$ cuboids fit in a $(a,b,c)$ box.
Second orientation the $n-1$ cuboids fit in a $(a-1,b-1,c-1)$ box.
Third orientation the $n-2$ cuboids fit in a $(a-2,b-2,c-2)$ box.
If the volume of one cuboid is $V$, the three orientations give $$nV = abc \\ (n-1)V = (a-1)(b-1)(c-1) \\ (n-2)V = (a-2)(b-2)(c-2)$$
Expanding the second and third, and substituting in the first, we get $$V = ab + ac + bc - a - b - c + 1 \\ V = ab + ac + bc - 2a - 2b - 2c + 4$$ so $$a + b + c = 3$$ and the $(a-2,b-2,c-2)$ box is going to have some negative dimensions.