Dimension of the sum of subspaces
This example will illustrate the difficulty.
Let $i,j,k$ be the standard basis vectors for 3-dimensional real space. Let $W,X,Y,Z$ be the subspaces spanned by $i,j,k,i+j$, respectively. Then $W+X+Y$ has dimension 3, $W+X+Z$ has dimension 2, but $W,X,Y,Z$ all have dimension 1, and any intersection of two or more has dimension 0. So, no formula using dimensions of the four spaces and their intersections can distinguish between $W+X+Y$ and $W+X+Z$.