Mentioning Specific Examples in the Motivation Letter

Do I have to mention specific described incidents (like the ones provided during the behavioral interview) in the motivation letter? In the letters I read, students said that their accomplishments were so and so. And they were parts of so and so. But they never mentioned a specific example like in the behavioral interview.
Your response would be highly appreciated. I am working on my cover letter and I do not know any one to help, moreover, I am new in the US so I do not know much about motivation letters, but I am very keen to give my best.
Many thanks.