One great example of a successful non-hierarchical structure is the Internet! The Internet is such a mess. It serves to show how this can be possible:

Applying the idea of networked thoughts to programming

The only way the internet is "organized" is by search engines. Google allows you to traverse the internet and search exactly what you're looking for based on associations and your search query.

Smart filtering to provide context

If search engines can provide this for the whole internet, there's no reason it can't be applied to our local computer's file structure. I think it would require a little bit more adjustment to make it suitable for personal user use, though.

For example, there would be a perpetual fear of "losing your files" or not being able to find them again without a container based file structure.