Overtraining has it's own set of symptoms which are easily identifiable. These include trouble sleeping, decreased immunity (presumably due to the decreased levels of glutamine in the body which also play a big role in supporting the immune system), irritability, loss of training motivation, lingering soreness, increased incidence of injury, etc. However, just experiencing ONE of these does not necessarily indicate that someone is technically overtraining. It cracks me up to see all these posts asking if people are overtraining or not. Overtraining is not always the culprit.
Usually, people get over-reaching and overtraining mixed up. Over-reaching is more or less training beyond your optimal training zone (in terms of training volume) in one or just a few workouts. You can over-reach but not overtrain. If you continue to over-reach, you will eventually overtrain and experience the aforementioned symptoms. To keep it simple...overtraining is more chronic in nature while overreaching can be acute. They are closely realted but not the same thing.
Now, to answer your question...you can flog a muscle to death but not necessarily overtrain your entire body. More than likley, that muscle group WILL NOT grow but you may not be overtraining that group to the point of causing your entire body to suffer the consequences. For example, it would be very possible for you to overtrain your legs, back, or chest to the point of sending cortisol levels sky high, thus affecting gains throughout your entire body. However, it would be harder to achieve this just by overtraing a smaller group like calves or forearms, simply because the strain it causes the nervous system to effectively train these muscles is much less and their catabolic influences on hormone levels is also much less.
I hope that answers you question without being too confusing, LOL!