I get this request a lot of times. Most organizations are flexible and will accommodate you. Some will look at this with suspicion that you are fishing for a better offer. So the question is, why do you need to extend? If you have a valid reason, there should not be a problem.
Another thing you can try is to join on the offered date and ask for a short vacation. Although most people are likely to deny such a request, if you can provide a good reason, you are likely to be told what's feasible. e.g. the HR may say you need to discuss with your manager, and they can connect you to him/her. As long as you can convince the manager with a reason, this may get approved.
That being said, there are times when this request may be declined. Some organizations have a pre-defined training plan for new grads which is quite expensive. Like you, there could be more hires and it may not be possible to move around those training's just to accommodate your late joining. Another possibility is that you are identified for a particular project / work. In that case, your boss may have to make a call. HR wont override business decision on the same.
If you are prepared for both scenario's, well there is no harm in asking. At most, someone is going to say no, you have to join on assigned date. Ask nicely, its worth a try :).