We investigate the frequentist coverage properties of credible sets resulting in from Gaussian process priors with squared exponential covariance kernel. First we show that by selecting the scaling hyper-parameter using the maximum marginal likelihood estimator in the (slightly modified) squared exponential covariance kernel the corresponding credible sets will provide overconfident, misleading uncertainty statements for a large, representative subclass of the functional parameters in context of the Gaussian white noise model. Then we show that by either blowing up the credible sets with a logarithmic factor or modifying the maximum marginal likelihood estimator with a logarithmic term one can get reliable uncertainty statement and adaptive size of the credible sets under some additional restriction. Finally we demonstrate on a numerical study that the derived negative and positive results extend beyond the Gaussian white noise model to the nonparametric regression and classification models for small sample sizes as well.