Modest Mouse/Tear Jerker

This indie-rock band sure knows how to open up the waterworks.

Little Motel. Though a sad song in its own right, the video to this song ramps it up to eleven -- with a mother watching her son die in a hospital, then carrying his body out to her car, driving him to a diner, a gas station, then putting him to bed at a motel, all played in reverse. Describing it in words makes it sounds creepy as hell. In fact, just watch it.
The World At Large, a soft, poignant song that documents confusion about one's place in life, breaks the heart of anyone who can relate, and a lot of people can relate.
The softly-sung, hopeless lyrics of Positive/Negative can strike particularly hard.
Night On The Sun, especially in the second half.

"There's one thing to know about this earth
We're put here just to make more dirt, and that's ok
It's alright... it's alright"

- Along with the naturally-Tear Jerker melody, this line is very, very powerful.
When paired with the video, Ocean Breathes Salty qualifies, for sure. While the song itself isn't particularly sad, one might find the music video to be strangely depressing (a little kid finds an injured bird, which is actually the lead singer in a costume, and takes care of it until it dies.)
Life Like Weeds
Wild Pack of Family Dogs
"Talking Shit About a Pretty Sunset" by Modest Mouse. The suicide references, weary acoustics, and achingly beautiful guitar/cello jam combine to make something even more raw and emotional then their "I'm utterly pissed at capitalism, romance, and the general public" noisefests.
In keeping with "Talking Shit...", "Bankrupt on Selling" may be one of the most depressingly cynical songs Modest Mouse has ever written, which is hard to do when you're a band that's widely known for being cynical.
The Moon and Antarctica has always been kind of a sad album, but "3rd Planet" can make some people start crying halfway through.
Then there is Missed the Boat.

"Oh and I know this of myself
We listen more to life's end gong
Than the sound of life's sweet bells..."

Float On. The song is all about how we have to keep on going, no matter how bad things get.
Trailer Trash, anyone?. I listened to this right after I watched Brokeback Mountain for the first time, and I'll be damned if it didn't just fit perfectly

"Short love with a long divorce
And a couple of kids of course
They don't mean anything
Live in trailers with no class
Goddamn I hope I can pass high school, means nothing"

There's something about the beautiful melancholy of Gravity Rides Everything that just tugs on this Troper's feelings. The realization of all the things in the world that weigh down on ones shoulders, and how there's nothing you can really do about it. It's an acceptance that's as comforting as it is hopelessly tragic.

"In the motions, and the things that you say
It all will fall, fall right into place
As fruit drops, flesh it sags
Everything will, fall right into place
When we die, some sink and some lay
But at least I... don't see you float away"

- The guitar spiraling away at the end is just icing on the cake