I'm pretty sure the answer to that is an emphatic yes: The Aharonov-Bohm effect has been proven many, many times experimentally. My recollection is that it is described as experimentally proven in the Feynman Lectures from way back in the late 1960s. Looking...
Ah, here it is: Volume II, Chapter 15, Section 15-5, "The vector potential and quantum mechanics" (starting on page 15-8), page 15-12, second paragraph from the bottom. The name is misspelled as "Aharanov" in some older editions, incidentally.
That reference likely not a proof, but I know I've seen other very specific, detailed, and far more recent papers and discussion of the effect, including some very nice ones using small lithographically etched circles. I liked that one especially because it make the idea of Aharonov-Bohm as a type of displacement vector easier to visualize. Looking again...
I don't believe this is the same paper I was thinking of, but it's 2008 and pretty similar in testing strategy. Definitely worth a look if you are interested, and likely has relevant references also:
Mesoscopic decoherence in Aharonov-Bohm rings, by A.E. Hansen, A. Kristensen, S. Pedersen, C.B. Sørensen, and P.E. Lindelof. The Niels Bohr Institute, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark
(February 1, 2008).
And two more for completeness! First, this most excellent paper found by @JoeHobbit (see his comment below) looks like it may be "the paper" that in 1986 -- much later than the Feynman mention -- first provided convincing proof of the existence of Aharonov-Bohm:
Experimental confirmation of Aharonov-Bohm effect using a toroidal magnetic field confined by a superconductor, by Nobuyuki Osakabe, Tsuyoshi Matsuda, Takeshi Kawasaki, Junji Endo, and Akira Tonomura. Phys. Rev. A 34, 815–822 (1986). This classic paper is also available here through scribd, and also here.
For a much more recent example of just how mature this area is these days, there is this 2013 (!) paper on using graphene (I love graphene) to explore Aharonov-Bohm oscillations:
Transport properties of two finite armchair graphene nanoribbons, by Luis Rosales and Jhon W. González. Nanoscale Research Letters 2013, 8:1 doi:10.1186/1556-276X-8-1 (2 January 2013).
(Kudos to Springer for making that and similar papers truly open, by the way!)
2013-02-18. Alas, I've not received any response from Dr Aharonov, though to be honest I thought it unlikely. Worth a try, though; I'd have loved to hear his view.
After going over Timothy Boyer's paper, and in particular after seeing that in 2006 he proposed some specific experiments (inaccessible without paying) to distinguish between relativistic effects and a true Aharonov-Bohm effect, my position is pretty simple: If his experimental setup looks reasonable, someone should should try Boyer's experiment and see what they get.
There is nothing unreasonable about the sequence of first noting possible modeling errors (which is all Boyer is really doing, and doing it pretty well from what I can see), next giving an alternative modeling option that addresses the concerns of the author, and then suggesting an actual experiment to distinguish between the predictions.
Why not just do the experiment? Seems like a good way either to firm up the nature of all Aharonov-Bohm results, or add some interesting new discussion if the experiment produces anything unexpected.
If Boyer's experiment is flawed in concept (again, I cannot see that paper!), can anyone on this group who has seen it say why?