Estimation of system and components reliabilities is considered when independent partition-based Dirichlet measure priors are assigned on component lifetime distribution. In our nonparametric Bayesian approach we assign independent partition-based Dirichlet measure (PBDM) priors, on jth component distribution function F_j,j = 1,2,..,K, with the parameter ?_j being a non-null finite measure on R+. Given the components right-censored lifetime data, we derive the nonparametric Bayes estimator of component reliabilities and an estimator of the system reliability function. The nonparametric estimator of system reliability function presented in Doss et al.(1989) is a special case of our estimator, obtained by letting ?_j(R)?0. Through simulation studies, we demonstrate that the nonparametric PL-type estimator has smaller bias, but higher root-mean-squared errors (RMSE) than our proposed estimator. Even when the prior mean functions do not coincide with the true distribution functions, the Bayes estimator has smaller or approximately equal RMSE than the nonparametric PL-type estimator. The proposed estimators are illustrated using a synthetic data set for a five component parallel system.