A molecular-level insight into the phase transformations is of great demand for the molecular systems for which the phase transition is a key process from the point of view of practical applications, organic phase-change materials being one of the most prominent examples. Such a detailed insight can be gained through cooling-rate computer simulations, i.e. simulations in which cooling is applied to a system at a constant rate. However, the ability of the simulations to give an accurate description of the phase transition process is an open question, since the cooling rates accessible in atomic-scale computer simulations are many orders of magnitude larger than those in experiments. In this paper we present the first computational study that systematically explores as to how the cooling rate affects the crystallization of organic phase-change materials such as paraffins. To this end, we performed a series of atomistic molecular dynamics simulations in which the cooling rate was varied over four orders of magnitude. Our computational results clearly show that a certain threshold in the values of cooling rates exists. When cooling is performed with the rates smaller than the threshold, computer simulations are able to qualitatively reproduce the crystallization process in paraffins. This includes an abrupt change in the temperature dependence of the density, enthalpy, and thermal conductivity upon crystallization. The crystallization enthalpy for these cooling rates was found to agree quantitatively with experiment. However, beyond this threshold, when cooling is too fast, the paraffin’s properties in simulations start to deviate considerably from experimental data: the faster the cooling, the larger part of the system is trapped in the super-cooled liquid state. As a result, one can observe a systematic decrease in the fraction of crystalline domains in a paraffin sample with cooling rate, leading eventually to a complete lack of crystallization at low temperatures. Both the crystallization temperature and the crystallinity decrease with cooling rate in line with experimental data. All the above strongly supports the conclusion that a proper choice of a cooling rate is of tremendous importance in computer simulations of organic phase-change materials such as paraffins. Cooling with the rates that are too high can lead to a completely inaccurate description of the crystallization process, as well as the thermophysical and structural properties.