Univariate Lévy processes have become quite common in the reliability literature for modeling accumulative deterioration. In case of correlated deterioration indicators, several possibilities have been suggested for modeling their dependence. The point of this paper is the study of three different dependence models: use of a regular copula, superposition of independent univariate Lévy processes and use of a Lévy copula. The three methods are first presented and analysed. In this way, it is shown that the multivariate process constructed through an ordinary copula cannot have independent increments in general, that is, it is not a Lévy process. The impact of a wrong choice for the model is next explored, based on data simulated from one model and next adjusted to all three models. It is shown that a wrong model can lead to either overestimate or underestimate the reliability function, which could be problematic in an application context.