I have encountered a similar problem and searched to find only this post relevant, although my polygons are simply two dimensional. Here is my import configuration.
Importing with configuration: sz_div_st_cell, public, geom,
F:\path\to_polygons.shp, mode=c, dump=1, simple=0,
geography=0, index=1, shape=1, srid=0
Shapefile type: Polygon
PostGIS type: MULTIPOLYGON[2]
Shapefile import failed.
After checking the input shapefile several times, I found that the polygon shapefile itself, which came from an postgresql export and underwent some operations in ArcGIS, had a field named 'geom' in its attribute table (.dbf), and caused the import failure. After removing that 'geom' field, which was the default geometry field name in shp2pgsql GUI, the import went smoothly and successfully.
Vector > Geometries Tools > Multipart to Singleparts
from the QGIS menubar and see if you can import again.