"Down In A Hole" is a kind of depressing only they can pull off; whether the tears come or not will depend on who you are, but hearing the singer accept that his life is over and it's all his fault evokes a certain gloom and sorrow rarely pulled of tastefully.
"Would?" might be even more depressing than "Down in a Hole".
The normal version is depressing enough, but the acoustic version is almost painful to listen to (and even harder to watch when you can see Layne's rotting teeth, constant shaking, and pained face). Hearing that song coming from a dying man is one of the most depressing things you'll ever hear.
"Nutshell". Good God, just..."Nutshell". To clarify, it was written by Layne Staley - enough said.
And "Rooster" as well. Especially the "Send me pictures of my boy" line... The "oohs" at the beginning are enough to tell that it is a depressing song.
Although, let's put it this way: said song might be the happiest on the album Dirt, and the subject is the guitarist's Vietnam vet father, and it's still talking about him getting shot at, and all his friends getting gunned down or dying from malaria.
"Whale & Wasp", which has no lyrics — but still manages to be...depressing as hell.
"Don't Follow", especially the part where Layne yells that he's scared to death, can be a real trigger. It's like listening to a best friend cry or something.
"Hate To Feel" - especially when you compare it to "Real Thing", which was about how much he wants his friends and family to back off about his addiction, even though he knows he needs help. "Hate To Feel" is basically a sequel, about how the heroin has taken over his life, and now he needs it because life is physically too painful without it.
"Black Gives Way to Blue", which is about the late Layne Staley.
Hell, you could say that the Self-Titled Album is all one big Tear Jerker - being written by Layne Staley. With the exception of four songs, they are about how he's all alone with no one willing to help and him unwilling to accept help with his addiction. It's truly depressing stuff, but it gets worse. He didn't remember writing those songs, because he was high when he wrote them.
Especially in the chorus of "Frogs", where Layne repeatedly asks, "Why's it have to be this way?", and the spoken word coda, where he looses such wonderfully hateful lines as "Off the wall I scrape you" and "Never gonna fuck with me again".
"Died" and "Get Born Again", the two final songs recorded with Staley before he died, especially the former song. Read to the lyrics to it, and you can almost feel his pain. Keep in mind he died less than 4 years after it was recorded. Also, read the aforementioned Wikipedia sections to hear how he looked during the recording. Oh, lord...
The music video to "Hollow" is nothing short of heartbreaking, especially with how slow and calculated the video shows the astronaut's demise.