In 1969, Seymour Papert and Marvin Minsky showed that Perceptrons could not learn the XOR function.
This was solved by the backpropagation network with at least one hidden layer. This type of network can learn the XOR function.
I believe I was once taught that every problem that could be learned by a backpropagation neural network with multiple hidden layers, could also be learned by a backpropagation neural network with a single hidden layer. (Although possibly a nonlinear activation function was required).
However, it is unclear to me what the limits are to backpropagation neural networks themselves. Which patterns cannot be learned by a neural network trained with gradient descent and backpropagation?