$geoIntersects will show these two polygons as intersecting:

{ "type" : "Polygon", "coordinates" : [[[ 1.0,  3.0], [ 12.0,  3.0], [ 12.0,  8.0],  [ 1.0,  8.0], [ 1.0,  3.0]]] },

{ "type" : "Polygon", "coordinates" : [ [ [ 6, 9 ], [ 7, 9 ], [ 7, 8.02 ], [ 6, 8.02 ], [ 6, 9 ] ] ] }

The top of the first polygon is .02 degrees away from the bottom of the second one. The range of false intersections increases with lattitude. I wasn't able to reproduce this with longitudinally near polygons.