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We exhibit examples of high-dimensional unimodal posterior distributions arising in nonlinear regression models with Gaussian process priors for which Markov chain Monte Carlo (MCMC) methods can take an exponential run-time to enter the regions where the bulk of the posterior measure concentrates. Our results apply to worst-case initialized ('cold start') algorithms that are local in the sense that their step sizes cannot be too large on average. The counter-examples hold for general MCMC schemes based on gradient or random walk steps, and the theory is illustrated for Metropolis–Hastings adjusted methods such as preconditioned Crank–Nicolson and Metropolis-adjusted Langevin algorithm. This article is part of the theme issue 'Bayesian inference: challenges, perspectives, and prospects'. [ABSTRACT FROM AUTHOR]