The quantum approximate optimization algorithm (QAOA) is the most studied gate-based variational quantum algorithm today. We train QAOA one layer at a time to maximize overlap with an n qubit target state. Doing so we discovered that such training always saturates - called training saturation - at some depth p∗, meaning that past a certain depth, overlap cannot be improved by adding subsequent layers. We formulate necessary conditions for saturation. Numerically, we find layerwise QAOA reaches its maximum overlap at depth p∗=n for the problem of state preparation. The addition of coherent dephasing errors to training removes saturation, recovering robustness to layerwise training. This study sheds new light on the performance limitations and prospects of QAOA.