The first time I got depressed after seeing an anime was Fate/Stay Night's last episode
But after watching 4-5 shows where the romance remains unfulfilled I became used to it. The other anime that really got me depressed was Elfen Lied - no need to explain why. And the third and final time that I got depressed was after watching Grave of the Fireflies - man that was the most depressing thing I've ever seen.