The assumption (scary word, I know) is that with SWIM you are saving JT files for all of the components that make up an assembly and then opening up the BVR of the assembly in Structure Manager to veiw an assembly of JT files that follow your revision rules, allow control of surpressed instances, and also follow the exclude from BOM option in Solidworks.
If you also create an assembly level JT, then you may see double JT files in Structure Manager.
Not completely clear on what you are trying to do with the JT. Import into another NX assembly or visualize in Teamcenter.
" I am trying to work in a multi-CAD environment where I have NX top levels that consume both NX data and Solidworks generated JT data."
This is exactly what we've been doing at Siemens Molecular Imaging for years going back to 2008.
We have no issues opening Solidworks assemblies in TC/NX via multi-CAD other than that you can only see the "Default" Solidworks Assy Config / TC BVR Arrangement because SWIM doesn't create BVR Arrangements for every Solidworks Assy configuration, but that rant is for another time.
You'll want to ensure that you follow all instructions on configuring Multi-CAD in both NX and TC. You are apparently missing something in your setup. Since you can view them in TC Vis, it's likely an NX Customer Default, Preference, or environment variable issue.