The Weibull distribution is one of the most used tools in reliability analysis. In this paper, assuming a Bayesian approach, we propose necessary and sufficient conditions to verify when improper priors lead to proper posteriors for the parameters of the Weibull distribution in the presence of complete or right-censored data. Additionally, we proposed sufficient conditions to verify if the obtained posterior moments are finite. These results can be achieved by checking the behavior of the improper priors, which are applied in different objective priors to illustrate the usefulness of the new results. As an application of our theorem, we prove that if the improper prior leads to a proper posterior, the posterior mean, as well as other higher moments of the scale parameter, are not finite and, therefore, should not be used.