System Information
Affects all operating systems. Tested on Linux AMD64, Windows 7 x64, Windows 8.0 x64.
Affects all graphics cards. Tested on an NVidia GeForce 760 GTX.

Blender Version
Working: without proxies up to 2.72b.
Broken: with proxies at least since 2.70
Broken: completely since 2.73

Short description of error
For larger animation projects (video games, machinima, etc.) it is useful to make use of Blender's library linking. Typically like this:

• dude.blend (contains Armature + meshes)
• - dude.jumps.blend (contains animations, links Armature from dude.blend w/ proxy)
• - dude.combat.blend (contains animations, links Armature from dude.blend w/ proxy)

Exporting animations from dude.jumps.blend or dude.combat.blend produces an FBX either not containing those animations or with detached animation tracks (checked in Unity 5.4, Unreal Engine 4.11, Autodesk FBX Review 1.4)

Blender 2.72b without proxy: export and animations working
Blender 2.72b with proxy: export contains no animations
Blender 2.77 SVN without proxy: export contains a single empty animation
Blender 2.77 SVN with proxy: export contains unworkable animations

Exact steps for others to reproduce the error

1. Create a mesh and animate it using bones and the Armature modifier, save to .blend file
2. Create a second .blend file and link to the mesh + Armature from the .blend file in step 1.
3. In the second .blend file, create a proxy of the Armature so you can animate it (space, 'Make Proxy')
4. In the second .blend file, create a new Action in the Dope Sheet and animate the proxy
5. Export the animations in the second .blend file using the current FBX exporter

I have uploaded example files allowing you to skip steps 1-4 here:
http://blog.nuclex-games.com/linked-armature-fbx-export-repro-case.zip