Matt Austern, the chair of the C++ standardization committee’s library working group, explained this decision of the committee in his Dr. Dobb’s article by historical reasons:
We discovered, with more testing, that even the [simple] example didn’t work with every STL implementation. In the end, it all seemed too murky and too poorly understood; the standardization committee didn’t think there was any choice except to say that STL containers aren’t supposed to work with incomplete types. For good measure, we applied that prohibition to the rest of the standard library too.
My understanding of this is that the committee did not want to invalidate existing implementations of the library by requiring them to support incomplete types retroactively.
In the same article he concedes that
In a future revision of C++, it might make sense to relax the restriction on instantiating standard library templates with incomplete types.
Given that the article dates back to 2002, and the prohibition remains in place in the current standard, I think that the decision of the boost designers not to wait for the future and build their own containers that allow incomplete types was fully justified.
Edit: See this answer for information on using incomplete types allowed by C++17 standard for some containers in the Standard C++ Library.