The periodic table tells us that there are 6 protons in a carbon atom. Is there a way to verify this first-hand? Or are we just expected to believe it unquestioned?
1) While not easy, it is possible to obtain mass-spectrum of full range of ions, including fully ionized. By number of peaks with Z/E corresponding as X, X/2, X/3 ... X/n it is possible to ensure that an element has exactly n electrons and protons. It is still difficult to ensure full ionization of heavier atoms, so the method is not applicable for heavier elements. This, however, is the only direct method I can think of. Should work for carbon, though.
2) Various X-ray-derived spectral methods. While not providing direct evidence, they do provide information on electronic shell structure, that can be compared with theoretical calculations.