When a function $\mathcal{f}$ : $\mathbb{R}$d → $\mathbb{R}$ is observed at points x1,...,xn, Gaussian Process (GP) interpolation is a standard method to infer the value of f at a new point x. However, what degree of trust can be placed in a prediction from a GP model? The natural way to deal with this question is to study the prediction intervals built from the posterior distribution of the GP model at x. These prediction intervals may be too optimistic or too pessimistic compared to the actual observations (in the precise sense described below). These considerations are crucial in the industry since prediction intervals that are too optimistic can be misleading and have disastrous effects. In contrast, overly pessimistic prediction intervals leads the user to increase the number of observations, resulting in an increasing approximation cost.