What Monte Carlo methods cannot do

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Possibility theory and statistical reasoning (2006, methodology): “Monte-Carlo methods applied to the calculation of functions of random variables cannot account for all types of uncertainty propagation [48].”
Constructing Probability Boxes and Dempster-Shafer Structures (2003, background): “3; Berger 1985) and the analytical techniques for probability boxes (Ferson 1996; 2002; Berleant 1996).”
Monte-Carlo based uncertainty analysis: Sampling efficiency and sampling convergence (2013, methodology): “In this class, Monte-Carlo based uncertainty analysis is taken to be the most dependable technique, as it is widely applicable—certain guidelines and limitations are to be considered however [5] [6] [7].”
Quantification of model uncertainty in RANS simulations: A review (2018, background): “Examples include imprecise probability theory [33], probability bounds analysis [34–36], Dempster–Shafer evidence theory [37], fuzzy sets [38], and credal sets [39].”
Dependence in Dempster-Shafer theory and probability bounds analysis* (2004, background): “The appearance of repeated parameters in expressions is a well-known problem with interval arithmetic and, indeed, with all common uncertainty calculi (e.g., Moore 1966; Manes 1982; Hailperin 1986; Ferson 1996a).”