I don't know what the multipart-complete file is there for. But have a look at this:
http://www.3dbenchy.com/3dbenchy-for-du ... -printing/
The one you're trying to print is all of those 17 objects together in one file. But that
means it has a lot of walls very close to each other while the simple STL file just has
"outer borders". Basically you're trying to print each part on it's own - but all at once
and very, very close together.
Have you tried the Slicing Behavoir options? Creating a single solid model should
slice properly.