Well, that and the fact that filesharing is very close in concept to voluntary botnets.
We've all (well, most) seen articles describing some of the bigger botnets as some of the most powerful supercomputers on Earth. If you could break a neural-net down into discrete components (and you can, with some types of interconnectivity, like low synapse-density connections between high synapse-density clusters) you could build a quite large distributed net.
Program this neural net framework in generic C for high portability and you could have many devices being part of a big brain in their spare cycles. Port a compatible version to CUDA and watch those graphics chips shine in a familiar element. The ability of multiple instances of the program to communicate with each other (even if crudely at first, like over the localhost interface) takes care of any multi-threading or heterogeneous processor issues.
I don't think it would be too hard to build for a skilled programmer with even a basic understanding of neural nets, the biggest issue I see about the whole idea (besides, of course, getting it popular enough to be useful) would be to find a good way to balance between waiting for synapse data from a remote machine and firing without complete data, which may be a long time coming if the remote machine is being used for another purpose right now. I guess it's like designing a brain wherein any part may just shut off at random times, and (possibly) re-awaken later. I suppose with enough training (and enough nodes) the network itself could work out a redundancy path, but that would increase training time by a human-scale amount, meaning that the things WE do (like breaking network cables, saturating our bandwidth or processors with higher priority tasks, etc.) will determine how fast this thing can learn how to organize itself.
If built to establish its own redundancy the poor thing probably wouldn't be able to learn anything for years or longer, due to our randomness of resource allocation. After it did work out a scheme we could probably learn a lot about ourselves by studying its network structure, but I digress...
I think it's a neat idea with a couple of large hurdles, but it could work if people were patient enough.