In this presentation, we will discuss issues pertaining to the construction of prediction intervals in the presence of model biases or discrepancies. We will illustrate this in the context of models for smart material systems but the issues are relevant for a range of physical and biological models. In many cases, model discrepancies are neglected during Bayesian model calibration. However, this can yield nonphysical parameter values for applications in which the effects of unmodeled dynamics are significant. It can also produce prediction intervals that are inaccurate in the sense that they do not include the correct percentage of future experimental or numerical model responses. This problem is especially pronounced when making extrapolatory predictions such as predicting in time. In this presentation, we will discuss techniques to quantify model discrepancy terms in a manner that yields correct prediction intervals. We will also indicate open questions and future research directions.