In the present paper we investigate the capability of finite element methods to correctly reproduce the stability range of finite strain problems in the incompressible regime. To this end, we develop a numerical scheme, obtained combining a stream function formulation with an isogeometric NURBS approach, which is able to sharply estimate the stability limits of the continuous problem. Using such a method, we show a pair of benchmark problems on which various well-known finite element methods largely fail in approximating the correct stability range.