All the time and not all the time. I know that's confusing, but let's say I say or do something that hurts someone or they take it in a way I didn't mean. I did not intend the hurt and I don't apologize for what I said. I do however recognize that regardless of my intention, they were hurt by it and for that I am sorry. Then I don't get resentful cuz I'm not really sorry for what I said and they see that I care about their feelings. Intention is not where I was wrong, I somehow didn't communicate properly. I use this strategy a lot with students to mitigate misunderstandings. Esp. when they make a bad joke and then get all indignant that someone is judging them for making a bad joke. It recognizes both the intent which was harmless but helps them realize that even jokes can have bad consequences and at the end of the day what matters is that you hurt someone's feelings.

Then there are times I'll apologize and it's not specified for what. I may be apologizing for one thing, and the other person thinks it's for something else but it opens up communication and makes us both happy.