I will discuss the suitability of truncated Polynomial Chaos expansions (PCE) and truncated Gram-Charlier expansions (GrChE) as possible methods for uncertainty quantification in nonlinear systems with intermittency and positive Lyapunov exponents. Based on a simple, statistically exactly solvable non-linear and non-Gaussian test model, I will show in detail that methods exploiting truncated spectral expansions, be it PCE or GrChE, have significant limitations for uncertainty quantification in systems with intermittent instabilities or parametric uncertainties. Intermittency and fat-tailed probability densities are hallmark features of the inertial and dissipation ranges of turbulence and it turns out that in such important dynamical regimes PCE performs, at best, similarly to the vastly simpler Gaussian moment closure technique. Moreover, I will show that the non-realizability of the GrChE approximations is linked to the onset of intermittency in the dynamics and it is frequently accompanied by an erroneous blow-up of the second-order statistics at short times.