Acoustic models used for statistical parametric speech synthesis typically incorporate many modelling assumptions. It is an open question to what extent these assumptions limit the naturalness of synthesised speech. To investigate this question, we recorded a speech corpus where each prompt was read aloud multiple times. By combining speech parameter trajectories extracted from different repetitions, we were able to quantify the perceptual effects of certain commonly used modelling assumptions. Subjective listening tests show that taking the source and filter parameters to be conditionally independent, or using diagonal covariance matrices, significantly limits the naturalness that can be achieved. Our experimental results also demonstrate the shortcomings of mean-based parameter generation.