Is there a $4$-component link such that upon removing any one of them you get the Borromean link? Is there a $4$-component link such that upon removing any one of them you get the Borromean link?
I've managed to get close but not quite. What I have gets me something similar to the Borromean link but two of the components actually form what I think is the Whitehead link.

You can sort of think of the Borromean rings as lying on three faces of a tetrahedron, with the center triangle of the usual presentation at a vertex.  By adding a component corresponding to the fourth face in a way so that the link has tetrahedral symmetry, you get the following:

While the outer component looks funny, it is the same as any of the other three, in the sense that there is an isotopy of the link sending this diagram to itself, but moving an inner component to the outer component.
