Large number of layers in a neural network?
In principle, there is no limit on the number of hidden layers that can be used in an artificial neural network.  Such networks can be trained using "stacking" or other techniques from the deep learning literature.  Yes, you could have 1000 layers, though I don't know if you'd get much benefit: in deep learning I've more typically seen somewhere between 1-20 hidden layers, rather than 1000 hidden layers.  In practice the number of layers is based upon pragmatic concerns, e.g., what will lead to good accuracy with reasonable training time and without overfitting.

Don't confuse artificial neural networks with the brain.  The behavior of artificial neural networks doesn't necessarily tell us anything about the brain, and you can build artificial neural networks whose structure doesn't match anything found in the brain.