Japan has strange customs that can often be baffling to westerners, so imagine going to a place where you just cry, yes you do nothing but bawl your eyes out until you run dry.
Well this practice is common and is referred to as Rui-Katsu; coined by a divorce counsellor, groups of people will gather to deliberately watch sad emotional short films to induce crying, the custom is said to relieve stress and create happiness, he first tried it out in his divorce counselling, not only did it work, it stopped a few divorces as well.
In regards to anime only one comes to mind that relates to this.

Yes, Air, the visual novel series by Key infamous for being one of the saddest anime of all time. You only have to look at this character to know what happens. Now I'm not a fan of Key as pretty much 90% of their work involves tears at some point and crying is not my first choice for entertainment but I do give Air credit for being a rui-katsu film, it's always going to make you cry and that's okay.
There's no shame in letting it all out, it may even save someone, after all we lose less people to mental health problems if we released our emotions more.