One-dimensional steady-state passage of direct electric current from a binary electrolyte solution into a charge-selective solid such as a metal electrode or an ion exchange membrane is hydrodynamically unstable. Instability is preceded by concentration polarization, i.e., depletion of the electrolyte in the interface diffusion layer and it yields a microvortical flow in this layer. An ambiguity persists regarding the mechanism of this instability. The buoyant mechanisms are disregarded because instability also occurs in a gravitationally stable position and in diffusion layers too thin for buoyancy to mediate the flow. Therefore, instability is attributed to the electric forces acting in or near the interface electric double layer. These forces cause a sliplike flow known as electro-osmosis, which comes in two varieties. One is the classical equilibrium electro-osmosis related to the space charge of the electric double layer. The other is the nonequilibrium electro-osmosis related to the extended space charge that forms near the interface at high depletion. Both types of electro-osmosis may yield instability. The question is which one is at work in each particular system. The nonequilibrium electro-osmotic instability, unlike the equilibrium one, is of the short-wave type. This implies that its induced vortices are small compared to the width of the diffusion layer. Therefore, this width, which has not been clearly defined in most experiments so far, is crucial for the identification of the instability mechanism. In this paper, we report the results of our combined experimental and theoretical study of concentration polarization in a custom-designed experimental cell with a particular cation exchange membrane. As a part of our study, we investigate the recently predicted thermoelectroconvective instability. This instability is of the long-wave type and its related flow involves a pair of wide vortices spanning the diffusion layer. We experimentally retrieve this flow, which clearly marks the width of the diffusion layer. We observe that for high voltages this thermoelectroconvection is accompanied by electro-osmotic instability. Upon the background set by thermoelectroconvection, we are able to conclude that the observed electro-osmotic instability is of the short-wave type and is thus due to the nonequilibrium electro-osmosis. We suggest that a similar approach might be useful for identifying the instability mechanism in other charge-selective systems as well.