Measurement error is non-negligible and crucial in SHM data analysis. In many applications of SHM, measurement errors are statistically correlated in space and/or in time for data from sensor networks. Existing works solely consider spatial correlation for measurement error. When both spatial and temporal correlation are considered simultaneously, the existing works collapse, as they do not possess a suitable form describing spatially and temporally correlated measurement error. In order to tackle this burden, this paper generalizes the form of correlated measurement error from spatial correlation only or temporal correlation only to spatial-temporal correlation. A new form of spatial-temporal correlation and the corresponding likelihood function are proposed, and multiple candidate model classes for the measurement error are constructed, including no correlation, spatial correlation, temporal correlation, and the proposed spatial-temporal correlation. Bayesian system identification is conducted to achieve not only the posterior probability density function (PDF) for the model parameters, but also the posterior probability of each candidate model class for selecting the most suitable/plausible model class for the measurement error. Examples are presented with applications to model updating and modal frequency prediction under varying environmental conditions, ensuring the necessity of considering correlated measurement error and the capability of the proposed Bayesian system identification in the uncertainty quantification at the parameter and model levels.