This is a rather hyperbolic statement. You concede edge cases, but the fluidity of phenotype means that it fails to map cleanly onto neatly delineated categories. We can look at the recent book describing the history of 'passing' reviewed here [1] for one of many examples.
>I haven't read much on the subject, but the idea that races can't be separated based on genetics is either false, or due to the fact that we don't have enough sensitivity in the measurement.
Again, I'd caution against letting your intuition be your guide on this topic. A cursory examination of the literature (i.e., 15 minutes on Google Scholar [2]) might lead you to the conclusion that: (1) genotype != phenotype, and (2) while we can reliably sort humans by ancestry using genetic information, there is variation within these clusters because the proportion of any individuals ancestry that arises from different populations also varies.
I think that the upshot is that the science is less useful when applied to a nebulous term like race, which involves a host of cultural, political and historical factors. The way we Americans define race in debates like these is a legacy of our binary 'one-drop rule' concepts of identity, and it isn't at all clear that appeals to science couched in sweeping generalizations have any connection to the biology.
[1]: http://www.nytimes.com/2011/02/27/books/review/Arsenault-t.h...
[2]: http://scholar.google.com/scholar?hl=en&q=genomics+race