When are these "Hit the daily reputation cap on X days" badges assigned? Are they assigned when somebody's reputation always grown in X days, or something else?
They are awarded when your daily reputation exceeds 200 points on 50/150 days. If you visit https://stackoverflow.com/reputation and scroll all the way to the bottom, you'll see the last two lines look like this:
rep cap was reached on x days rep cap was exceeded on y days
I remember reading an answer from Jeff saying the second number applies to those badges.