We consider two dependent random times T and U, that correspond to two successive events. This setting is motivated by an application to insurance subscription, where a potential dependence exists between a time before effectiveness of the contract T, and a time U before its termination by the policyholder. The setting also extends to various types of applications involving two duration variables with some hierarchical link between the events. Indeed, since a contract can be terminated only after it becomes effective, data are subject to a particular type of censoring, where the variable U is systematically censored when the variable T is. In this framework, a nonparametric conditional copula model is considered, in the spirit of (Gijbels, Veraverbeke, & Omelka, 2011). The uniform consistency of the conditional association parameter is obtained under conditions of dependence structure and of censoring mechanism. A simulation study and a real data application show the practical behavior of the method.