We report increases of cosmic-ray intensity at low L shells, on the ground as well as on the altitude of 500 km, during the selected intervals when magnetosphere was strongly disturbed. The geomagnetic transmissivity for cosmic ray vertical access has been computed for four events, using two geomagnetic field models. The first event, namely, dated November 20-22, 2003, led to an increase of galactic cosmic-ray intensity due to the improvement of magnetospheric transmissivity at neutron monitors with high nominal cut-off rigidity. Other two events with high-energy solar proton emissions, namely, dated October 28 and November 2, 2003, caused different responses at middle latitudes. The first one followed by the strong geomagnetic disturbance led to the shift of the penetration boundary of protons, having (50-90) MeV at CORONAS-F satellite to L < 3, coinciding approximately with the cut-off reduction expected by Dst depression, while the other one, without remarkable Dst decrease, did not shift the outer boundary of penetration below L approximate to 4. And the fourth of the events, on November 8, 2004, with strong geomagnetic disturbance, yielded a complicated structure of cosmic-ray time profiles: superimposed on the cosmic-ray decrease, viewed by neutron monitors and by CORONAS-F at high latitude; an increase of intensity at middle latitudes both on the ground and on the altitude of 500 km has been observed during the Dst depression.