I've heard of this concept of 'set-based' filesystems (although with different terminology), and I think it's brilliant. And not just for documents either. Generally speaking, websites have moved away from storing data in a hierarchical matter (e.g. old-school Internet directories), and on to flat, tag-oriented categorization (e.g. delicious) because of the abundance of information to categorize. Hierarchies do not scale and are difficult to change, whereas free-form categorization do not suffer from this.
Personal systems have more files on them than ever before, and fitting them to hierarchies that were invented decades ago can be tricky. These issues have been abated for now with things like OS X's Spotlight and Win7's Libraries, but my suspicion is that set-based filesystems would be far simpler than the current paradigm.
A good example off the top of my head is resolving what command-line executables are available. In most operating systems, this is done by setting a PATH variable with a list of directories that contain the executables. While this is trivial for the type of people on HN, it is far from for your average user. Instead, command-line executables could be given a specific tag, and the command-line would be able to execute any application that has that tag.
Now, whether set-based filesystems could replace hierarchical ones is a different question. Such a filesystem would diverge so greatly from current assumptions that people might be too aversive to change. It wouldn't be the first time; I think Plan 9 was light years ahead in so many respects, but it never took off because it was such a radical departure.