Can atomic partial charges be measured in molecules experimentally?
Let me put it this way. There are no atoms in molecules. There is just a continuous cloud of electron density, and as you correctly pointed out, we have no way to pinpoint where one atom ends and another starts. Or rather, we have many ways to do that, all slightly different and all inherently arbitrary. (My personal favorite is the concept of Bader charges, but that's a matter of taste.)

So yes, the very idea of partial atomic charges is unavoidably vague.