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Yes, they do forgive. Mostly pre-screen offers I've been seeing.
I don't know if this example counts, but I'll tell you anyway. A young friend of mine got a Capital One card with a promise of a CLI after 5 on-time payments. He went over his limit once and Capital One sent a letter saying they will still honor his CLI if he still makes 2 additional payments on-time on top of the original 5 for a total of 7 billing cycles. Last week, he got a CLI exactly as promised. I don't think my friend knows his credit because he has none, but Cap One definitely is helping him build credit and establish a good relationship. Go Cap One!