MakerBot Print is our newest print-prepration software, which supports native CAD files and STL assemblies,
allows you to interact with all your printers via the Cloud, and many other exciting new features.
Solved myself(thanks, Google!)
Reason: S3D sees self intersecting surfaces (Repair - Identify self intersecting surfaces. It highlights the very edges I had problem with)
Solution: Mesh - Separate connected surfaces. It splits the model as several lillte models, but at least now it is sliced as expected