No, I find that the words "I am sorry" are bandied about carelessly. Yes, we are supposed to forgive someone when they are sorry. However, saying I am sorry and understanding what to be sorry for and really feeling it are two different things.
Also, saying "I am sorry" and then shortly after doing the same thing again that the person apologized for in the first place is abusing the phrase, in my opinion. Saying "I am sorry you feel that way" is just as bad. To me that translates to something like "I am right, and it's your fault that you are feeling disappointed or hurt by my action."
I'd rather have a person think about something for a while, then explain why they are sorry and then apologize. That beats a quick, careless "sorry," anytime. That, in my opinion, is the key to the offended party to really get over the issue, instead of dragging it out again and again .