The Gamma-ray Burst Monitor consists of 12 sodium iodide and 2 bismuth germanate scintillation detectors designed to enhance the scientific returns from Fermi in the study of γ -ray bursts. It has a fixed non-paralyzing dead time of 2.604 μ s for each detected event except for those in the overflow channel of the 12 bit analog to digital converter of each detector, which have a longer dead time of 10.42 μ s. At very high counting rates the pulse pile-up effect in the detectors leads to spectral distortions and also to additional event losses. These effects are relevant for the spectral analysis of high-flux astrophysical events such as gamma-ray bursts, soft gamma repeaters and solar flares, as well as terrestrial gamma-ray flashes. In this paper we present the results of post-launch tests using engineering detectors and equipment on the ground, along with detailed simulations of these effects at very high counting rates. The simulations enable us to assess the qualitative and quantitative changes in spectral shapes and event losses at different count rates. The observed spectra at various count rates using high intensity radioactive sources such as Cs 137 and Co 60 agree well with the simulated spectra. We also re-test the analytical model developed previously to correct for the pulse pile-up effects in GBM detectors. We also study the expected spectral distortions of specific spectral models commonly used in GRB analysis, including the Band model and power-law models with and without a high-energy cutoff.