No. What’s happened is that you added new cubes while the first (and then all the others) were still in edit mode. So they are now all one object. The right way is to add a cube, hit TAB to go into Object mode, move your 3D-cursor to where you want the 2nd cube to be then add it.
You can seperate your cubes by selecting only 1 vert of i cube, hold your mousecursor over that cube and press L (link) or Ctrl-L and it will select all the verts of the cube (and leave the verts of the other cubes unselected). Now press P (sePerate) and you have a seperate object.
Now, with that cube selected and in Object mode hit M (move) and choose which layer.