The drilling process is considered one of the most expensive and complex in the oil and gas industry. This paper is focused on the dynamics of a drill-string in which stick–slip oscillations can happen, reducing operational performance and increasing the costs. The main concern of this work is to analyze the consequences of a passage from a soft to a harder rock layer in the system response and how uncertainties influence this response. A one degree-of-freedom model is proposed where the top speed is prescribed, the drilling mud is represented by a viscous friction and the bit–rock interaction is represented by a non-linear model. First, the response of the system is analyzed when there is no rock transition, i.e., a single rock layer is considered. The transition from a soft to a harder rock layer is then included and the deterministic and stochastic responses of the system are assessed. The results show that different scenarios can significantly change system response.