Which experiments prove atomic theory? Which experiments prove atomic theory?
Sub-atomic theories:


*

*atoms have: nuclei; electrons; protons; and neutrons.

*That the number of electrons atoms have determines their relationship with other atoms.  

*That the atom is the smallest elemental unit of matter - that we can't continue to divide atoms into anything smaller and have them retain the characteristics of the parent element.

*That everything is made of atoms.  
These sub-theories might spur more thoughts of individual experiments that prove individual sub-atomic theories (my guess is more was able to be proven after more experiments followed).
 A: I once heard Uhlenbeck give a lecture on this to high school students over the Christmas break at the Rockefeller Univ. years ago.   He recounted a published argument he attributed to Einstein around 1905 (I think), which was that atoms were real if you could count the number of them/mole (Avogadro's number) many different independent ways, and you always got more or less the same answer.   So Brownian motion, gas law, counting with an atomic force microscope, X-ray diffraction, spreading oil film, and many other possibilities would all count as subarguments to the main argument, i.e. that atoms were real.   If anyone knows the reference(s), I would appreciate them.
This was a live question at the time.  For example Mach, who died in 1916, was apparently an atomic skeptic.
A: No experiments prove any theory. Experiments can only refute theories.
A: The history of atoms is definitely intertwined with quantum mechanics. There are many features of the quantum theory that make atomic nature of our world apparent. But here I'd like to state an earlier result.
Thomson's 1897 discovery of the electron not only showed that atoms exist but also that they have substructure.
A: I would say that one experiment that demonstrates the atomic nature of things is the observation of Brownian motion. But it is not the experiment itself that convinces that things are made of atoms, rather its theoretical explanation given by Einstein in one of his 1905 papers (actually Einsteins work for his PhD was on the subject of atomic theory and there are several publications in the period 1903-1905). Of course there is also the observation of Rayleigh who calculated Avogadro’s number by the distance from which he could make out the figure of Mount Everest, assuming that light is scattered by atoms and that is why far away objects look fuzzy (1,2). Also scattering experiments demonstrated the atomic nature of things.
(1) Rayleigh, On the transmission of light through an atmosphere containing small
particles in suspension, in Scientific Papers by Lord Rayleigh Vol. 4, pp. 247–405,
New York: Dover, 1899/1964.
(2) P. Pesic, Eur. J. Phys. 26, 183 (2005).
(3) Patterson, G. Jean Perrin and the triumph of the atomic doctrine (2007) Endeavour, 31 (2), pp. 50-53.
A: I think that the points made about Einstein's theoretical explanation for the observed Brownian motion and the observed Perrin experiments on it are quite valid.  But perhaps one could quibble that actually the forces on the pollen were produced by molecules...not by atoms... and perhaps one could resist the point by what is more than a quibble: it proved the reality of things that were too small to be seen, on the scale of atoms, but atomic theory is a little more than that.
Rutherford's alpha particle scattering experiments played a major role, too, besides giving the idea of atomic structure (even though it is called in the O.P. sub-atomic, which is true).  The combination of Rutherford and Rayleigh and Einstein--Perrin and the Millikan oil-drop experiment might be the best experimental verification of atomic theory.  After all, an entire theory needs several reinforcing experiments about quite a variety of phenomena to really support it, a point which was also made by Einstein as quoted in the answer by Mr. Goldberg.  
A: Answering the question requires some understanding of the ongoing controversies in the Philosophy of Science. Ontology refers to the existence of objects and effects, and Epistemology refers to human knowledge - how can we come to know about the existence of objects and effects. For unobservable objects and effects, almost by definition, there can ONLY be indirect evidence for their existence (such as Brownian motion for molecules). "Realist" Philosophies of science argue that such indirect evidence is enough for us to deduce (pragmatically) the existence of unobservable objects. "Empiricist" Philosophies say that science does not depend on existence or nonexistence of unobservables. According to these empiricist philosophies, the question of "whether or not atoms exist" can never be settled, and hence it does not matter for science. 
The philosophy of science called "Critical Realism" developed by Bhaskar Roy offers substantial clarity on how philosophy matters for the question under consideration, which must be unpacked into two different questions. The ontological question "Do atoms exist (as part of external reality)?" will have the same answer regardless of whether or not there are human beings around to ask this question. The epistemological question is: how can we learn about the existence of atoms (especially since they are unobservable to our five senses)? The answer now depends on our human senses, experiences, logic, theories, etc., and not just on external reality. A crucial distinction is between "certain knowledge" and "uncertain knowledge". If we ask: "Can we be certain that atoms exist?", the answer must be NO. Atoms are a theory about external reality which provide an explanation of a diverse set of phenomena. This provides strong indirect evidence for their existence. Nonetheless, it is always possible that tomorrow a different theory will come along which can explain all of these phenomena. For example, if string theory is true, then what appears to be an atom is really something else, which manifests as an atom at the scales that we can observe. Even more radical theories may be true which dispense with matter altogether. 
One school of philosophy holds that knowledge must be certain. If we accept this, then we can never have knowledge that atoms exist. The Pragmatic philosophers, on the other hand, allow for some uncertainty in knowledge. Taking a pragmatic stance, at the moment, existence of atoms seems to be the best hypothesis which allows us to explain a myriad of phenomena, and is coherent with a lot of other theories which are widely accepted - this coherence also provides additional indirect evidence for existence.  
