Is the power of the "Magic Words" dead?

I generally am a nice person but at times tend to be a touch egoistic in my approach (a flaw that I know most humans share, whether they agree or not). However once the rush of blood passes, I always make it a point to offer my apologies first, most people are in the same boat as me and "accept my" / "offer their" apologies and we generally have a laugh about it. There are the few pricks that take this as a sign of cowardice and try to drive home their point.

My question is, does the "Magic Words : Sorry and Please" still hold their value? Also if you could recount some incidents to highlight the same, that would reaffirm my faith. My very first incident was "When I was asked to back down from a fellow competitor (after having him pinned down), it was during my martial arts training and can still feel the Master's cane when I refused to listen". This happened when I was in my third grade and still holds me in good steed.