We extend the results of Yu et al. (2015b) of the novel sharpness angle measurement to a large number of spectra obtained from the Fermi gamma-ray burst monitor. The sharpness angle is compared to the values obtained from various representative emission models: blackbody, single-electron synchrotron, synchrotron emission from a Maxwellian or power-law electron distribution. It is found that more than 91% of the high temporally and spectrally resolved spectra are inconsistent with any kind of optically thin synchrotron emission model alone. It is also found that the limiting case, a single temperature Maxwellian synchrotron function, can only contribute up to 58+23 -18% of the peak flux. These results show that even the sharpest but non-realistic case, the single-electron synchrotron function, cannot explain a large fraction of the observed spectra. Since any combination of physically possible synchrotron spectra added together will always further broaden the spectrum, emission mechanisms other than optically thin synchrotron radiation are likely required in a full explanation of the spectral peaks or breaks of the GRB prompt emission phase.