It is important to admit when you’re wrong. Showing fallibility is a strength, not a weakness. No one is perfect. There is a sense of integrity and trustworthiness that is associated with a person that can be honest enough to admit errors. That being said, the words “I’m sorry” don’t always have the power to fix whatever mistake was made.
You can’t unsay or undo what has already been said or done. No matter how honorable or noble you are and no matter what the circumstances, everyone has their own personal standards. The lines that they have personally determined cannot be crossed if relations with them are going to continue. Perhaps, so long as you are willing to admit your fault, it can be worked out. But it is not usually forgotten. It may fade, over time, but it will always be there.
There is no easy fix. Similar to being unable to buy forgiveness, there comes a point when enough becomes enough. Words are our most valuable resource, but they are only as valuable as the people using them determine. If you keep making the same promise, but never deliver, your promises lose their value. The same goes for saying “I’m sorry” too many times. If you keep apologizing, especially if it is for the same thing, what value do you think gets placed on your apology?
Say the words too often, the words lose the little healing power they possess. While capable of fixing minor infractions, they add up. Eventually, the power wears off. The words get drowned out by how many times they have been used. It eventually becomes unclear whether you actually mean it, or use it as a crutch for flaws that you know you have but are incapable or unwilling to fix. They become irrelevant when the person receiving the apology loses faith in the value of the words and the source they are coming from.