While I was only speculating there, yes. Instead of having one parent field for each document, there would be another table, document_parent, with two fields: the document ID and the Id of another document to be its parent. You could have as many entries in the table for document #x as you needed to have parents. It’s the same as any "category" structure, where a catagory_item table allows items to belong to more than one category.
A menu snippet would show document #x in the submenus of as many folders and you put it, each one linked to the same document #x.
I believe that you could use TVs to do the same thing, although a menu snippet would have to be written that will make use of the TVs. That would be a better option, I think, than trying to completely rework the core "tree" structure as it is. There is something along these lines here:
http://modxcms.com/forums/index.php/topic,398.0.html
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