If quantum tunneling is possible, is there a maximum thickness of material a particle can go through, and is it random? I've read a few articles here and there about quantum tunneling, but this one left me with a few questions that I'm having trouble finding the answers to online. 
The article says that particles can pass through walls--this is something I've heard before--but does it matter how thick the wall is? Does the density of the wall matter? 
Previously before reading this article I had thought that quantum tunneling was completely random, and that there was no way to predict it, but it says "Previously, theoretical calculations had predicted the timing of quantum tunneling, but never before has it been directly measured with this accuracy." I'm not entirely sure that I'm interpreting this statement correctly. Does this mean that it's not completely random? 
I always imagined that it might be possible (but extremely unlikely) for the particles of one macroscopic object to kind of "line up," and travel through a solid object together. I know it's highly unlikely that all of the particles would do this in unison, but is it theoretically possible? 
 A: 
The article says that particles can pass through walls--this is something I've heard before--but does it matter how thick the wall is?

The term 'wall' here is used somewhat metaphorically. More precisely, mathematically we speak of a potential energy wall or barrier, see for instance in the diagram below:

A quantum particle with energy $E$ comes in from the left and meets a rectangular potential (energy) barrier (wall, if you prefer) of height $U_0>E$.
In Classical mechanics the particle would simply bounce off the barrier, unable to overcome or penetrate it due to too low energy. The barrier and everything to the right of it is thus the Classically forbidden zone.
But a quantum particle is different: it can penetrate and tunnel through the barrier to some extent.
In the figure, $\Psi_i$ and $\Psi_e$ represent the wave functions of the incident and exiting particle. In quantum mechanics the probability density of the particle is proportional to the modulus squared, $|\Psi|^2$, of amplitude of the wave function. As you can see, the amplitude diminishes as the particle tunnels through the barrier. Simply put this means that the probability of finding the particle in that region diminishes. So it's less likely to find it right of the wall than left of it. But Classically that probability of the former would be zero.
Quantum tunnelling is not random at all: the probability densities left, in the barrier and right of it can be calculated precisely. Both wall thickness and wall height affect the outcome in a predictable manner. There is no real maximum thickness: as thickness goes up probability of finding the particle right of the wall goes down exponentially until it is practically zero.
In the article you linked to the experimenters use a laser to lower the potential barrier caused by electrostatic attraction between the electron and the helium atom's nucleus, thus increasing the probability of finding the electron some distance away from the atom.
(Source of figure.)
A: There is not a maximum distance that can be tunneled through. 
More distance of materials means significantly lower probability of tunneling. But that probability never goes to 0. Of course, experimentally, we've only ever seen tunneling at small distances; tunneling can quickly get so improbable that in the lifetime of the universe, it would (on average) never occur.
Note: You may have seen problems with particles in a potential well, where the probability outside the well is 0. This is an entirely impossible scenario in real life. You cannot create a potential well with a perfectly vertical wall, as it corresponds to infinite force, which none of the known forces can physically provide.
A: Short answer: The range of tunneling (or, more technically, the probability to find the particle at a point 'behind' the potential barrier,) is not limited. However, the probability of finding the particle at that point 'behind' the potential barrier is lowered. That's dependent on the shape of the potential.
To find out: create your own potential barrier, and solve the (time independent) Schrödinger equation for the specified potential, invoke scattering, solve for the transmission and reflection coefficient and you will find that unless your potential barrier is infinitely high there is always a non-zero transmission coefficient T (and associated with a non-zero probability of finding the particle).
This is not really mathematically rigorous, but that's the 'visual' description of the theory. Gert's explanation offers a mathematically solid explanation.
Don't forget that if your hypothetical potential barrier is infinite there is no tunneling possible!
