Backward waves (wave with opposite signs of the phase and group velocities) in a circular cylindrical shell filled with an acoustic fluid are studied. Three theories are applied to describe the motions of the shell: Kirchhoff-Love theory, refined shell theory with modified inertia, and 3D theory of elasticity. Numerical results for combinations of parameters ensuring the existence of backward waves are presented. By comparison of these results, the limits of applicability of the shell theories are investigated. It is shown that the using of two-terms operator of transverse inertia is sufficient to extend the applicability of the shell theory up to high-frequency range.