If one hits the daily cap, then decides to sleep so they can do it all again the next day; then someone removes a vote
/fill in the blank, does the attained limit still count or is it only if you end the day on the +200?
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The same way that you don't get more reputation even if you are below 200 reputation (if someone removes a vote or accepted answer), the cap is already reached for the day.
Once a badge is awarded it is not retracted, but if it is later invalidated somehow (such as from the retraction of a vote that would have caused you to hit the rep recap), you won't earn another badge for the same thing when you meet those conditions again.
That isn't relevant for Mortarboard, since you can only earn one once, but it would count for the counter towards Legendary or Epic.
Edit: if the rep is removed on the same day, then you wouldn't get the badge, because the rep cap badges are not calculated until EOD. (They are not awarded instantly on first hitting the rep cap earlier in the day -- you will not have counted as hitting the cap unless you are there at EOD, when the batch process to calculate these badges is run.)