That short answer is very misleading. Those 6 layers are histological layers of density and structure in the cerebral cortex (eg how the cross sections look a little different). You might have many (30+) layers of cells with high interconnectivity in each of those 6 layers. Current estimate is 100 billion for the number of neurons, but the estimate of inter-connectivity is on the order of 100 trillion synapses. The human brain complexity many orders of magnitude higher than anything we have ever "simulated" in silicon.
The brain's layers are both much more loose in their allowed interconnections but also more modular in their functionality. There are different visual areas (usually 5) at different locations of the neocortex that connect to each other (feedforward and back) in complex ways. As residents of the neocortex, each visual area will thus have 6 layers which are in turn dividable into sublayers (often 1 - 4). These layers are structural but are also partially organizational tools for the scientists.
Mammalian brains are so densely packed that we don't really know what their internal architecture actually looks like--that's why estimates of the number of neurons keeps increasing over time [1]. Most of our models are based on some autopsies, and rely on things like fMRI or electroencephalography (not often you get to spell that word!) to work out which portions of the brain are involved in certain tasks. We have no hope of being able to build a connectivity graph between neurons at our current state of technology.
The short answer is that the cerebral cortex is usually identified as having 6 layers. These can't be correlated with layers in a neural network, however. I do seem to recall a spitball number of 30-ish in relation to the visual cortex, but I've been unable to find any references as to what that number would actually represent, let alone if it's been superseded.
[1] The current estimate is around 100 billion. If you want to know the number involved in, say, the visual cortex, that appears to be on the order of a few billion.