What does double dissociation really tell us? In psychological research, a single dissociation is when a manipulation leaves one cognitive function (say, A) intact whilst severing another (say, B). This indicates the functions A and B are at least partially independent. A double dissociation is when there exists, in addition, a manipulation that does the reverse (i.e. leave B intact whilst severing A). On the Wikipedia page for dissociation in neuropsychology/cognitive neuroscience (I'm assuming the "manipulation" is a brain lesion now) it says that "establishing a single dissociation between two functions provides limited and potentially misleading information, whereas a double dissociation can conclusively demonstrate that the two functions are localized in different areas of the brain." This can't really be true, can it? Consider the following Venn diagram, where the (necessary) neural substrate of A and B are represented, including their intersection: 



A double dissociation could consist of a manipulation in A \ B and another in B \ A. The functions would still be partially dependent, however, and not localized in completely different areas of the brain. So, in effect, the only thing a single dissociation proves is that the circles for A and B are not identical, and the only thing a double dissociation proves (in addition to the fact that they're not identical) is that one is not a subset of the other. Is that about right?
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.