It's not about the design of the network, it's about the training. Optimizing the weights of a DNN is a non convex optimization problem solved using gradient based search for weights that minimize a loss function. If you can only search in 10 dimensions, it's harder to move in the correct direction. Once you're searching in 1000 directions, you can usually find a path that gets you to the right place. Think of it this way: there may exist a 10 dimensional solution, but it requires learning a very dense solution.