The article you site refers to the Bulgarian method, which produces world class lifters who can handle the high intensity, constant training, and become very strong because of it. But propenents tend to not mention all of the people who are spit out the back of the Bulgarian system because their bodies can not handle the strain.
It is the people that can not handle constant training that can over train, and that is the majority of the population. Over training is exacerbated by two things especially:
Bad form: You see this in baseball pitchers. They throw hard for months, but bad form combined with too many pitches leads to injury. These are not necessarily instantaneous injuries, but often injuries related to tearing and inflammation, which could have been avoided with fewer pitches, more rest, or especially, better form.
Overloading the eccentric movements: Eccentric movements occur when a muscle is extended as it is trying to contract. For example, when lowering a weight to the floor, fighting gravity, your bicep is contracting, but the weight is extending the muscle at the same time. This movement can cause substantial gains in strength, but excessive weight, or excessive numbers of movements can cause long term injuries such as tennis elbow.
My guess is that the person profiled in the article above is used to the average person leaving his gym, while the genetic anomalies stay and successfully work within his program.
Added comments: It is my opinion that non-injury symptoms of over training are really caused by under recovery. In other words, you could potentially deal with the problem by sleeping more, or eating better, rather than necessarily reducing the workout.