Kerr frequency comb generation relies on dedicated waveguide platforms that are optimized toward ultralow loss while offering comparatively limited functionality restricted to passive building blocks. In contrast to that, the silicon-photonic platform offers a highly developed portfolio of high-performance devices, but suffers from strong two-photon absorption (TPA) and subsequent free-carrier absorption (FCA) at near-infrared telecommunication wavelengths, thereby rendering Kerr comb generation a challenge. Here we present a model to investigate the impact of TPA and FCA on Kerr comb formation. Our model combines a modified version of the Lugiato-Lefever equation with a refined relation to precisely describe the fast space and time dependence of the free-carrier concentration along the circumference of the microresonator. Using this refined model, we derive conditions for modulation instability, in particular for necessary pump powers depending on TPA parameters and free-carrier lifetimes. We validate our analytical predictions by time integration and study the impact of fast free-carrier dynamics on Kerr comb formation. We find that silicon microresonators may be suitable for Kerr comb generation in the NIR, provided that the dwell time of the TPA-generated free carriers in the waveguide core is reduced by a reverse-biased p-i-n-junction and that the pump parameters are chosen appropriately.