Magnetization dynamics measurements at subnanosecond time scale have been performed on Fe-Co-N high-saturation soft magnetic films with Permalloy nanolayer seeds and having a saturation magnetization of 1.9 MA/m (1900 emu/cm3). The damping parameter α varies from 0.011 to 0.018, depending on applied bias field. The peak frequencies of the imaginary permeability spectra and the zero-crossing frequencies of the real permeability spectra cannot be fitted with a fixed value of anisotropy field, indicating that the Kittel equation is only qualitatively valid for these films. A phenomenological damping criterion is established based on a small-signal solution of the Landau-Lifshitz-Gilbert equation: critical damping occurs in a soft magnetic film when the damping parameter is α=2/√(χ0), where χ0 is the initial permeability. The experimentally observed damping parameters are smaller than the critical value for the range of bias fields employed. The Fe-Co-N-based films have a ferromagnetic resonance frequency of 2 GHz at zero-bias field, showing great promise for applications in write heads and integrated inductors in a frequency range >1 GHz.