Electrochemical models are a widespread framework to simulate lithium-ion (Li-ion) batteries and to design their battery management algorithms. To obtain numerically efficient models, the polynomial approximation (PA) is one of many order reduction techniques applied to the solid-state lithium-ion (SSLi+) diffusion equation, the core of the so-called pseudo-two-dimensional electrochemical model (P2DM). Although the validity of PA is constrained to slow-varying current scenarios, many algorithms reported in the literature to estimate the state-of-charge of Li-ion cells rely on low-order PA-based dynamic models (PADMs) derived from the P2DM. Moreover, assuming that most properties of their low-order counterparts are inherited, some authors suggest that PADMs of arbitrary high order can be used to provide very accurate approximations of the state-of-charge, even under complex operating conditions. Nevertheless, to authors knowledge, in the open literature there is not a proper analysis to support such assertion. In this paper, by introducing a systematic method to derive PADMs of arbitrary order, the ability of high-order PADMs to reproduce the average and surface concentrations described by the SSLi+ diffusion equation is investigated in time and frequency domains, with the aid of classic control systems theory techniques. The main result shows that PADMs of order greater than 2 are structurally fragile as non-minimum-phase zeros as well as spurious unstable modes are induced by the PA technique, implying that high-order PADMs are not suitable for simulation or estimation purposes because of their weak internal structure. Graphic abstract