We demonstrate the existence of spurious modes in finite elements with incompatible modes when a geometrically nonlinear displacement analysis with small displacements and strains is performed. The spurious modes are a direct consequence of the incompatibility of the elements with displacement boundary conditions. We derive a critical compressive strain condition analytically, and show that the critical strain can be quite small, with small displacements, if the geometric aspect ratio of the elements is large but still practical. In numerical examples we give further insight and results in correspondence with the analytical theory, and demonstrate that spurious modes can be triggered in practical small strain analyses when using elements with incompatible modes.