Models on the 3D Warehouse import just fine. Aside from these, I find it to be quite often that freebies downloaded from other corners of the 'net - especially as .FBX or .OBJ - have broken textures. It gets old, real fast, having to fix these textures manually! It feels like almost half of the free models I download (outside of 3D Warehouse) have major texture issues.
The obvious solution is to avoid using the limited and obsolete formats such models were exported to as an intermediary for use in other software and use these models in their native format (usually .MAX, as in 3DS Max, or .blend for Blender). But 3DXchange does not support those.
Another solution would be to use .glTF 2.0...
Q: Will a future version of 3DXchange support the import and/or export of the new .glTF format? This format was initially released by the Khronos Group in October of 2015. And it is under development to this day.
This format is rapidly being adopted by different companies, sites and software. It's gaining in popularity due to it's features and being so modern. Among other features, it boasts support of Physically Based Rendering shaders, sparse accessors and morph targets for techniques such as facial animation. It also has schema tweaks and breaking changes for corner cases or performance such as replacing top-level glTF object properties with arrays for faster index-based access.
In 2017 Microsoft announced that they will be using glTF as their 3D asset format across their product line. The popular Sketchfab 3D model and art platform has built-in glTF support. (All downloads on Sketchfab provide .glTF as an option through the site's automatic conversion.)
Other companies and products that support this format include (among others) Blender, UX3D, Facebook, Oculus, Google, Adobe, Box, TurboSquid, and Unreal Engine. The format has been noted as an important standard for augmented reality, integrating with modeling software such as Autodesk Maya, Autodesk 3ds Max, and Poly. (Also, there is ongoing work towards import and export in Unity and an integrated multi-engine viewer / validator.)