I don't know about the rest of the world but here in Germany Prof. Günter Nimtz is (in)famous about his display experiments that he claims show that quantum mechanical tunneling happens instantaneously rather than according to Einstein causality. In the past, he got a lot of publicity for that and according to Heise online he has at least a new press release.
All these experiments are similar: First of all, he is not doing any quantum mechanical experiments but uses the fact that the Schrödinger equation and the wave equation share similarities. And as we know, in vacuum, Maxwell's equations imply the wave eqaution, so he uses (classical) microwaves as they are much easier to produce than matter waves of quantum mechanics.
So what he does is to send a pulse these microwaves through a region where "classically" the waves are forbidden meaning that they do not oscillate but decay exponentially. Typically this is a waveguide with diameter smaller than the wavelength.
Then he measures what comes out at the other side of the wave guide. This is another pulse of microwave which is of course much weaker so needs to be amplified. Then he measures the time difference between the maximum of the weaker pulse and the maximum of the full pulse when the obstruction is removed. What he finds is that the weak pulse has its maximum earlier than the unobstructed pulse and he interprets that as that the pulse has travelled through the obstruction at a speed greater than the speed of light.
Anybody with a decent education will of course immediately object that the microwaves propagate (even in the waveguide) according to Maxwell's equations which have special relativity build in. Thus, unless you show that Maxwell's equations do not hold anymore (which Nimtz of course does not claim) you will never be able to violate Einstein causality.
For people who are less susceptible to such formal arguments, I have written a little programm that demonstrates what is going on. The result of this programm is this little movie.
The programm simulates the free 2+1 dimensional scalar field (of course again obeying the wave equation) with Dirichlet boundary conditions in a certain box that is similar to the waveguide: At first, the field is zero everywhere in the strip-like domain. Then the field on the upper boundary starts to oscillate with a sine wave and indeed the field propagates into the strip. The frequency is chosen such that that wave can in fact propagate in the strip.
(These are frames 10, 100, and 130 of the movie, further down are 170, 210, and 290.) About in the middle the strip narrows like in the waveguide. You can see the blob of field in fact enters the narrower region but dies down pretty quickly. In order to see anything, in the display (like for Nimtz) in the lower half of the picture I amplify the field by a factor of 1000. After the obstruction ends, the field again propagates as in the upper bit.
What this movie definitely shows is that the front of the wave (and this is what you would use to transmit any information) everywhere travels at the same speed (that if light). All what happens is that the narrow bit acts like a high pass filter: What comes out undisturbed is in fact just the first bit of the pulse that more or less by accident has the same shape as a scaled down version of the original pulse. So if you are comparing the timing of the maxima you are comparing different things.
Rather, the proper thing to compare would be the timing when the field first gets above a certain level, one that is actually reached by the weakend pulse. Then you would find that the speed of propagation is the same independant of the obstruction being there or not.
Update: Links updated DAMTP-->IUB