I think there are some pros and cons to this. Everyone has done a great job of pointing out the pros, most of which I do agree with.
However, what your original instructor may have been warning you of is the possibility of failure in your QA/QC duties if the 'other' duties become too overwhelming.
For example, when I moved into the QA Manager role, that was my only role. After awhile, I had several more roles added. At first, it was working great, but due to (in my opinion) incompetence in some of the other management roles, it has become overwhelming.
I basically have three jobs - QA Manager, front line supervisor (for inspection staff) and a company 'executive' which requires me to spend about half of my day in meetings.
What I have found wearing these three hats, my meetings are mandatory and take roughly 50% of my day. We have a lot of issues in production at the moment that the (in my opinion) incompetent managers are failing to address. So, when I am released from a meeting, I have at least three people who had called and are waiting on me. As I walk to the floor to address these three questions, I get bombarded with several more questions. This takes up the other half of my day.
What this means is that my QA/QC responsibilities are suffering and our QMS is not being managed appropriately. So, when it comes time for our re-cert audit, I will be isolated in a conference room for about three weeks to get us 'ready'.
This is not an effective QMS, and I believe this is what your instructor is warning of. If you can avoid these issues, I don't see any problem with accepting extra responsibilities, and as others have pointed out, it gives you more value come hard times.