While FBX files can have multiple animation takes embedded in a single file, 2 of the most popular DCC tools, Maya and 3DS Max do not ship with the functionality to export the Animation Take data. These tools have a single animation timeline, and export the animation data a single take.
Artists desiring to specify multiple animations relating to a single model or scene have a few options, but all of them essentially defer defining this data to further down the asset pipeline.
The typical pipeline workflows are:
Export each separate animation into a separate FBX file. There are a whole bunch of problems with this idea.
Export modelling data to Autodesk MotionBuilder (previously called FilmBox, the origin of the FBX format) or another equivalent tool and use these to define the desired takes. These will import cleanly into separate takes.
Max and Maya have a paid plugin (fairly inexpensive - $9 USD on TurboSquid as of the time of writing) allowing the artist to define multiple takes from the Maya and Max animation timeline. These are fairly simple tools, just defining a portion of the timeline to be each take, but are sufficient for most purposes.
Define all "takes" on a single timeline (with spacers between the desired takes) and export it as is. Use tools from the target middleware (Storyboard Designer, in our case), if they exist, to "slice" the animation into separate animations.
In order to support workflow 4, we would have to support the concept of slicing/splitting the incoming animations. As of Storyboard 4.2, this functionality is not supported.