Previously, Bayesian anomaly was reported for estimating reliability when subsystem failure data and system failure data were obtained from the same time period. As a result, a practical method for mitigating Bayesian anomaly was developed. In the first part of this paper, however, we show that the Bayesian anomaly can be avoided as long as the same failure information is incorporated in the model. In the second part of this paper, we consider a problem of estimating the Bayesian reliability when the failure count data on subsystems and systems are obtained from the same time period. We show that Bayesian anomaly does not exist when using the multinomial distribution with the Dirichlet prior distribution. A numerical example is given to compare the proposed method with the previous methods.