They can, actually, but not the ones you'd usually expect. They're not big, tragic tearjerkers but they are hugely poignant. And they happen to be my top four favourites.
The Good, the Bad and the Ugly has at least three genuine tearjerker moments for me. One is where the protagonist, Tuco, the so-called "Ugly", finds out about the deaths of his parents and gets into a fight with his brother; one is where he gets beaten by a nasty soldier courtesy of "The Bad". And the third is where he's forced to stand on a cross with a noose around his neck and is screaming for mercy. Even though I know he lives, the scene is still really suspenseful and I start sobbing madly, every time.
Seven Samurai has a few as well. The scene where Kikuchiyo slams the farmers and the samurai. But that's more out of empathy for him. And another moment where he tells Kambei that his family was killed by bandits, just like the child he was holding. And then he gets jealous of Kyuzo, and as he tries to prove himself, he gets several of his allies killed in the process.
The Host has the scene where Gang-du loses it in front of two doctors when they ask him why he didn't tell the authorities that his daughter was alive. There's also the moment where his father Hee-bong tries to defend his son in front of his siblings. The desperation in his voice is really powerful.
High Noon has a moment that gets me choked up, when Marshal Kane's last deputy deserts him and he collapses onto his desk in rage and despair.
It's not one of my top movies, but there's a scene in The Hobbit: An Unexpected Journey that got me quite a bit, when it seemed like Bilbo was lost forever in the mountains as he and the dwarves were crossing them.