Figuring out how to scope a site is full of so many nuances that it doesn't really lend itself to broad generalizations. It takes an understanding of the community and community building to know how broadly you can scope a subject before you start alienating the participants.
In the broadest sense, a new proposal isn't generally considered a duplicate of an existing site if there's a substantial body of questions that isn't already well-supported on another site. Splitting off a site like Mathematica from Stack Overflow was in recognition of the large body of content that went well beyond the programming questions of Stack Overflow. There was a very large and enthusiastic group of users who didn't consider themselves "Stack Overflow programmers." In other words, the subject of "Mathematica" wasn't being very well-served by its former home.
The Area 51 FAQ talks about Should my idea be part of an existing site, or its own site? But that only considers the mechanics of where a question might best fit. As it turns out, there's also a very real human element of forming communities that has to consider if a group would even recognize a broad subject as a place they would be interested in. Avid chess players aren't likely to embrace a site labeled "for boardgames"; poker interests go beyond "a card game"; and you can't start herding pet owners, parents, and amateur astronomers into sites like biology, cognitive sciences, and physics. You can argue the validity of that intuition, but those are the type of arguments we consider every day.
Modeling all of our sites to be as big and inclusive as Stack Overflow was a nice sentiment — but carving out an overly-broad swathe in the name of "a larger community" isn't going to be of much use if the target audience overlooks the site looking for more-focused interests. Sometimes a series of smaller, more-focused sites end up being greater than the whole. Then again, you can't splinter off every subject and brand into vanity sites, either. And you can't simply start combining smaller, vaguely-related subjects into one space to create the appearance of a larger community to get a site created. Splitting or combining subjects can really make or break a site, but doing so for the wrong reasons is ultimately self-defeating.