"Responsibility is one of those things that you actually have to take. The key with actually getting more responsibility is to take it. Not to ask for it, but to take it. You know, each and every day, when you see an opportunity to step into a situation that exceeds your current job description, do it. Dip your toe into it. The concern that a boss has in giving you more responsibility is that you're not up for it, and so if you actually start to take responsibility, slowly but surely you're going to be proving to your boss that you're up for the challenge.
So one of the things that you need to be cognizant of in taking responsibility, is not stepping on someone else's toes. You need to be aware of what your colleagues' jurisdiction is, and actually ask the question versus just taking on the task. Ask them if they need some help. Most people appreciate the initiative and appreciate the fact that you're willing to take on some of their work. So, you know, ask them whether or not this is possible. Take on the responsibility. You're not going to get more responsibility unless you go out and take the initiative.