I haven't really been brought down to tears by any anime thus far. I don't think I have literally cried in the last decade at all. At least I don't remember it.
But the following anime, off the top of my head, got me feeling very sad at times:
Mushishi
Monster
Mononoke (+Bakeneko from Ayakashi, and Bakeneko is probably the saddest arc in the show even though the others were quite sad too)
Honey and Clover (or, unrequited love: the anime lol ;_
Aoi Bungaku with its sad arcs (Ningen Shikkaku, Kokoro, Run Melos especially)
Welcome to the NHK
Jigoku Shoujo
Now and then, Here and there
Haibane Renmei
There are many other titles I want to mention but while they all have their moments, the number of shows that actually made me feel really really sad consistently is rather low. I second some of the mentions in the thread like 5cmps, Grave of the fireflies etc. too.
For the record, I don't find any of the titles mentioned in your opening post consistently or particularly sad. So you might wanna be cautious with the shows I mentioned because they are really sad, especially if you can relate to the content.