There are some specific situations where the 3ds Max FBX Plug-in does not export animation layers by default.
Animation that exists on multiple layers does not export correctly as the 3ds Max FBX Plug-in conditionally supports Biped animation layers.
As long as you are at the uppermost (top) layer, and use the Bake animation option in the FBX exporter, you can export all animation from the top down on the layers that reside beneath the top layer.
If you do not activate Bake Animation, only the top layer of animation is exported. This is because the 3ds Max FBX Plug-in Bake Animation option scans through any unsupported controllers in 3ds Max and then re-samples the animation for export.
