Packing 1x1x4 bricks into a 6x6x6 box

Is it possible to fit $53$ rectangular bricks, each $1\times1\times4$, inside a $6\times6\times6$ box?

No. Split the box into $27$ blocks of size $2\times2\times2$ and colour them black and white like a 3D chessboard: $14$ black blocks ($112$ cells) and $13$ white ($104$ cells). Any $1\times1\times4$ brick runs through two blocks of each colour, covering $2$ white cells, so at most $104/2=52$ bricks fit, even though $53\times4=212<216$.