The growing need for residential housing in Latin American countries has led to the construction of reinforced concrete buildings with wall thicknesses as low as 8–10 cm. Such walls have typically only a single layer of vertical rebars and are therefore particularly susceptible to out-of-plane failure. In order to investigate the response of the corresponding wall boundary elements, twelve reinforced concrete members with a single layer of vertical rebars were tested under tension–compression cycles. The objective was to gain insight into the parameters governing wall instability and out-of plane failure, namely the thickness, reinforcement ratio, and eccentricity of the longitudinal rebars with respect to the member axis. This paper summarises the results of the test program, where the specimens’ response is analysed also at the global and local levels. The results show that the crack pattern has an important influence on the out-of-plane behaviour and the conditions leading to out-of-plane failure are described. Furthermore, the differences between members with a single layer of vertical rebars and members with two layers are discussed. The influence of the parameters considered in the experimental program is addressed, showing that sections with small thickness and large reinforcement content are more prone to out-of-plane failures. Finally, predictions given by existing models are compared to the new experimental data. The entire data set is publicly available.