This paper discusses a nonparametric method for estimating under random censorship a hazard rate function with a possible change-point. First, motivated by a practical example, we focus on the situation where the location of the change-point is known. Within the general framework of counting processes, using the kernel method of estimation in partial linear models, consistent estimators for the rate of change at the breakpoint and for the resulting hazard rate function are obtained. The case of an unknown location is also addressed. The performances of the estimators in both models are checked via simulations.