I don't think an anime has ever had me flat-out cry before.
I have had knots in my throat and slightly teary eyes while watching some series, though.
I remember seeing the last two episodes of Narutaru (Jesus, a lot of unexpected people died in the last two eps), being the first tragedy anime I've seen, was almost too intense for me even if it was tame compared to other series.
I've watched more tragedy anime lately now, and the types of sadness that hit me the most are people suffering and dying or being raped as children.
Usually when children or teens have to go through things like that, even in fiction, you can't help but feel bad for them and have a knot in your throat or cry. ;-;