In fusion devices, the prediction of large magnetic island dynamics and saturation is of utmost importance for disruption control strategies. In experiments, this prediction can be provided by generalized Rutherford models. The purpose of this paper is to present an unified comparison between Rutherford models and first-principles simulations, across the various mechanisms leading to the growth of a magnetic island. More precisely, systematic comparisons of generalized Rutherford models with numerical 2D-reduced-magneto-hydrodynamic simulations have been done for a simple tearing mode and for neoclassical tearing mode (NTM) in presence or not of turbulence in order to assess the suitability of using generalized Rutherford models in experiments. It is found that agreement between simulation and model is not straightforward. For a simple tearing mode, prediction of saturated island size is possible only if the island is small enough. However, generalized Rutherford models fail to predict the whole island dynamics and as a consequence the saturation time. Predicting the behavior of NTMs is possible by using fitting parameters in generalized Rutherford models. Moreover, although the seeding mechanism is not taken into account in generalized Rutherford models, simulations show that NTM dynamics and saturation depend on the physical mechanisms at play to seed the island. In particular, a NTM presents different dynamics whether or not turbulence is present. Finally, it is shown that the amplification of a turbulence driven magnetic island to a NTM requires a bootstrap current that is strong enough.