I am not completely sure what you mean.
If we intersect two n-dimensional objects the cut/intersection can be of any dimension. What do you mean with "the whole thing"?
What we can say is that if we intersect an m-dim and an n-dim object the resulting dimension is at most min(m,n).
However in the theory of fractal and negative dimensions, they work with the probability of dimensions. Consider N-dim space with m-dim object A and n-dim object B then most probably is codim(AnB)=codim(A)+codim(B) where codim(X)=N-dim(X).
For example take two planes in 3d space the most probable cut is a line.
The codim of a plane is 3-2=1 and so the codim of the cut shall be 1+1 and indeed is this the codim of a line. Or cut of a plane with a line is most probably a point and indeed 3-2 + 3-1 = 3-0. Or a last example: the cut of a plane with a point should most probably be ... empty, but what dimension has the empty set? Lets compute: 3-2 + 3-0 = 3 - x. Hence x=-1 and this is where negative dimensions come into play.
Unfortunately in the moment I havent found a thourough introduction to this topic, but maybe have a look at
http://classes.yale.edu/Fractals/ especially the
page about dimension algebra. I think Manelbrot has introduced those probability and negative dimensions, at least he also has published about it.