There's an interesting caveat associated with this statement, which comes about when you ask: Does this mean that I could measure the kinetic energy of the particle and find it to be negative? The answer is no. But can I measure the position of the particle and find it to be in a region where the kinetic energy must "logically" be negative? Yes!
How do we reconcile these apparently inconsistent statements? The answer goes to the very heart of quantum mechanics. It turns out that energy and position are what we call noncommuting variables, in the sense that they cannot simultaneously be measured to arbitrary precision. Notice (1) that this statement does not say it is "very hard" to measure them to arbitrary precision (because, for example, as might be said classically, the system might be "chaotic" and we don't know the initial conditions or various influencing factors well enough); rather the statement says flatly that it is impossible, with any conceivable instrumentation, to make these two measurements perfectly precisely. Also notice that (2) the statement specifies "arbitrary" precision, that is, precision as good as we care to spend money and time making. In fact the statement does not rule out simultaneous measurements which may be by conventional standards very good, but not completely precise.
Quantum mechanics turns out to put definite limits on the minimum level of uncertainty in one member of a set of noncommuting variables given a certain level of uncertainty in the other. What this means in the case of classically forbidden regions is the following: if we measure the kinetic energy of a particle precisely enough to rule out the possibility that it might be positive in a given region -- that is, we establish a certain region as classically forbidden -- then the minimum level of uncertainty in a measurement of the position of the particle is such that we cannot be sure whether it is in this classically forbidden region or not! Conversely, if we measure the position of a particle precisely enough to know that the particle is definitely in some region, we cannot measure the kinetic energy of the particle precisely enough to determine whether this region is in fact classically forbidden.
To summarize: we cannot simultaneous know that a region is classically forbidden and that a particle is really located there. In a sense we can't "catch the particle in the act" of being quantum mechanical!
How do we know it is, then? Because, as discussed later on in the previous page, we can observe a consequence of the particle being able to penetrate a classically forbidden region -- its ability to appear on the other side of such a region, having "tunneled" its way out of prison walls made by classically forbidden regions. That is, we can measure the energy as definitely less than that required to get "over" such walls, and measure the position of the particle some time later as definitely outside the walls.
Now is that not mind-blowing?