Background: Two soft-photon amplitudes, the two-u-two-t special ( TuTts) amplitude and the Low amplitude, are known to produce quantitatively similar npy cross sections, but they predict quite different pp. cross sections for those kinematic conditions in which the nucleon scattering angles are small ( less than 25 degrees.). Purpose: These two amplitudes have been applied to systematically investigate three different nucleon-nucleon bremsstrahlung ( NN gamma) processes: pp gamma, np gamma, and nn gamma. The nn gamma process is explored for the first time. The primary focus of this work is to investigate the contribution of the proton and the neutron anomalous magnetic moments to all three NN gamma processes for projectile energies above 150 MeV and for laboratory scattering angles (theta(1) and theta(2)) lying between 8 degrees and 40 degrees. Method: A special soft-photon expansion in which the TuTts amplitude is expanded in terms of the Low amplitude plus additional amplitudes is utilized to explore the relationship between the TuTts and Low amplitudes and the reasons why they agree and disagree. We also used the TuTts amplitude to calculate the NN gamma cross section with and without the anomalous magnetic moment contributions to explore the importance of that element of the electromagnetic current. Results: The TuTts amplitude describes well the available pp. cross- section data. The anomalous magnetic moment contribution is ( i) significant in the pp. process when each scattering angle is less than 25 degrees but insignificant when each scattering angle is 40 degrees or greater and ( ii) insignificant in the np gamma process for all scattering angles. The nn. cross sections for the TuTts and Low amplitudes differ substantially for the kinematics investigated. Conclusions: In general, the Low amplitude agrees well with the TuTts amplitude when anomalous magnetic moment effects are not significant, but the two amplitudes can yield quite different predictions when such effects are significant. These findings have enhanced our understanding of the fundamental emission mechanism governing NN gamma processes, and they explain why the TuTts amplitude should be used to describe all three NN gamma processes.