Unfortunately not in FileMaker natively, however there is a plug-in by 360Works called SuperContainer that does allow you to create functionality like this.
Check it out at: http://www.360works.com/supercontainer/
I think it is more for moving and managing documents, but definitely worth looking into sa it may solve your query.
Troi File can do this, though it's kind of expensive. A somewhat clunky but free alternative on a Mac would be to create an AppleScript droplet, on your Desktop (or in the file's folder, if local) which could get the dropped file(s) paths, and tell FileMaker to add them to the current record (if files, then you'd have to add them to a linked table as new records).
A similar alternative would be to create a folder to drop items in, with a Folder Action attached to do the above.
I wrote a longer post, but the darn authentication time limit is too short, and it was lost.