This study addresses the performance of the extended and extended-cubature Kalman filters for state estimation of ill-conditioned stiff continuous-discrete stochastic systems. Stiff models arise naturally in practise as, for instance, the Van der Pol oscillator in electrical engineering and the Oregonator reaction in chemistry. Their accurate state estimation is a challenging task, which is made even more difficult by including ill-conditioned measurements, here. The authors' goal is to study accuracy of these methods when the Van der Pol oscillator and the Oregonator reaction become increasingly ill-conditioned. Thus, the authors determine efficient tools for estimation of stiff stochastic systems in the presence of round-off and other disturbances. They examine both conventional filters and their square-root forms. The square-root methods are obtained in two ways, namely, by solving the square-root moment differential equations and by square-rooting the filter itself. They prove that only the square-root filters grounded in the second approach (with use of stable orthogonal decompositions) are numerically stable and provide the highest state estimation accuracy in the ill-conditioned stiff stochastic examples under consideration.