What does double dissociation really tell us?
Double dissociation does not prove the independence of certain cognitive functions and their neural substrates. What it does is provide stronger evidence for which the best explanation is the independence of certain neurocognitive systems. 
The brain is on many aspects not a marvel of engineering with many areas being responsible for a spectrum of functions and behaviours or for different aspects of a seemingly unitary cognitive faculty. On top of that, it is often difficult to make a proper classification and distinction between certain functions and behaviours (examples are episodic/semantic memory and perceptual/semantic priming) which, in many cases, limits the validity of the conclusions drawn. 

So I don't know if a "conclusive demonstration" is possible. However, under properly defined terms, a double dissociation can provide strong support for independence with little possibility of an intersection between neurocognitive systems because if an intersection existed, it would be highly possible for a function to be retained at some level even after the hypothetical brain system responsible for it, suffers a lesion.