[Blizzer]

I would have to say the most tragic anime I have seen is Hanbun no Tsuki ga Noboru Sora,
I guarantee even if you are the stoniest person alive this anime will have some effect on you, it's a very bittersweet story.
If you haven't seen it, go watch it!

Other sad moments I particularly remember:

Spoiler for Utawarerumono:

When Aruruu was hysterically crying over the dead body of the old man, I shed a few tears, I think it was Aruruu's denial that made it so hard to watch.

Spoiler for Gundam 00:

When Saji visits Luoise in hospital and gives her the present of the rings, Louise cries and says she is unable to wear them and shows him her arm was severed, what made it so sad was Louise's pain and also Saji's painful reaction.