Dennett argues that human consciousness is essentially a single-threaded program laboriously constructed to run on an essentially parallel machine ("Consciousness explained"; I'm unfortunately still reading.)

We can, I believe, access the underlying machine. Processing social cues (cf. autists), playing Quake or protein folding is too hard for our single-threaded conscious brains, but clearly within human capacity. From personal experience, many high-level mathematicians (professors etc.) also have highly effective intuitions (of course, the ideas still have to be checked by the conscious mind - even highly effective intuitions produce quite a bit of nonsense, and almost always forget some details.)