Calculations of high-energy processes involving the production of a large number of particles in weakly-coupled quantum field theories have previously signaled the need for novel non-perturbative behavior or even new physical phenomena. In some scenarios, already tree-level computations may enter the regime of large-order perturbation theory and therefore require a careful investigation. We demonstrate that in scalar quantum field theories with a unique global minimum, where suitably resummed perturbative expansions are expected to capture all relevant physical effects, perturbation theory may still suffer from severe shortcomings in the high-energy regime. As an example, we consider the computation of multiparticle threshold amplitudes of the form $1 \to n$ in $\varphi^6$ theory with a positive mass term, and show that they violate unitarity of the quantum theory for large $n$, even after the resummation of all leading-$n$ quantum corrections. We further argue that this is a generic feature of scalar field theories with higher-order self-interactions beyond $\varphi^4$, thereby rendering the latter unique with respect to its high-energy behavior.