Sea ice has been suggested, based on simple models, to play an important role in past glacial–interglacial oscillations via the so-called “sea-ice switch” mechanism. An important requirement for this mechanism is that multiple sea-ice extents exist under the same land ice configuration. This hypothesis of multiple sea-ice extents is tested with a state-of-the-art ocean general circulation model coupled to an atmospheric energy–moisture-balance model. The model includes a dynamic-thermodynamic sea-ice module, has a realistic ocean configuration and bathymetry, and is forced by annual mean forcing. Several runs with two different land ice distributions represent present-day and cold-climate conditions. In each case the ocean model is initiated with both ice-free and fully ice-covered states. We find that the present-day runs converge approximately to the same sea-ice state for the northern hemisphere while for the southern hemisphere a difference in sea-ice extent of about three degrees in latitude between the different runs is observed. The cold climate runs lead to meridional sea-ice extents that are different by up to four degrees in latitude in both hemispheres. While approaching the final states, the model exhibits abrupt transitions from extended sea-ice states and weak meridional overturning circulation, to less extended sea ice and stronger meridional overturning circulation, and vice versa. These transitions are linked to temperature changes in the North Atlantic high-latitude deep water. Such abrupt changes may be associated with Dansgaard–Oeschger events, as proposed by previous studies. Although multiple sea ice states have been observed, the difference between these states is not large enough to provide a strong support for the sea-ice-switch mechanism.