You have to have a very specific culture in order to enable employees to feel as if it is acceptable to ask for more responsibility. It may not seem counter intuitive to you, but to an employee receiving a paycheck for a specific set of responsibilities, the cognative leap can sometimes be pretty large.

There are a number of reasons for this. Not wanting to appear dissatisfied is one. The false understanding that if a manager wanted them to step out into another area they would ask is another. Perhaps most difficult to overcome is the feeling that if you haven't been given more responsibility in an area, it is becasue your superiors do not see you as competent in that area.

There is a fairly well studied phenomena that highly skilled people don't necessarily feel that they are competent. Sometimes the fact that you have not be specifically asked to step into a roll is enough to confirm any suspicion that you may be less than competent in that role. Given that, it is very easy to fall into a "If they need me, they will ask" attitude.