The ring current is conventionally considered responsible for the shift of the boundary of solar proton penetration into the inner Earth’s magnetosphere during magnetic storms. The cases of a boundary shift were observed in some works on the dark side before the onset of a magnetic storm, i.e., at positive values of the Dst index. In this work, this type of shift of the penetration boundary is considered in detail with two storms as examples. It is shown that the corresponding distortion of the magnetosphere configuration is induced by an increase in the solar wind pressure during the initial phase of a magnetic storm. The current induced in this case on the magnetopause is closed by a current in the equator plane, which changes the configuration of the dark side of the inner magnetosphere, weakens the magnetic field, and allows solar protons to penetrate the inner magnetosphere. The significant difference in the positions of the penetration boundary and the boundary found from models of the magnetosphere magnetic field can be explained by insufficient consideration of closing currents.