I would like to elaborate on Hayley Sarg's point: your ability as an engineer is the sum of your skills and experience. Note that this experience can be gained at any point in your life, not just your job. If your hobby is building go-karts, chances are you will have a fairly extensive knowledge of vehicle dynamics. If you hobby is building RC aircraft, then chances are you have a fairly extensive knowledge of aerodynamics.
A "pigeon-holed" engineer who describes himself as such is an unhappy specialist. At some point, you will have to specialize in some pursuit of engineering (even if your specialization is mechanical integration, which deals with interfaces between a variety of subsystems). The moral of the story is that you should only specialize once you have found your area of interest.
The best way to do this is to find SMALL companies; the smaller the better. Responsibility usually goes hand in hand with an understanding of how the entire engineering system fits together; thus allowing you to develop an understanding for a variety of subsystems. As your understanding of each individual subsystem improves, so will the breadth of you knowledge. To broaden your skill set, you will have to work with each subsystem (this happens naturally in small companies).
Finally, you will learn far more about engineering from being given responsibility than you will by simply being delegated menial tasks. Furthermore, great engineers are those who seek challenges, not those who are content with repeating menial tasks that will one day be replaced by a software program.