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Tear Jerker: Hey Arnold!
The ending of "Pigeon Man". Pigeon Man has to leave, because his home was destroyed by Harold and his friends, but he thanks Arnold for teaching him that some people can be trusted, and he hopes that there'll be another Arnold wherever he ends up.
The life of Pigeon Man in general is sad. Heck, the original ending had the Pigeon Man kill himself by hurling himself off the building, but Nickelodeon said "No," as they felt it was too depressing (and didn't want to get blamed if a stupid or suicidal viewer actually thinks this is a good idea), so they replaced it with Pigeon Man using his pigeons to carry him off into the horizon.
Fridge Tear Jerker: He accomplishes that by having his pigeons grab onto rings on his outfit which are attached to thick tethers. He had those rings on his clothes from the start of the episode. From the beginning, he knew he would have to do this someday...
Helga revealing how much her family life sucks on "Helga on the Couch." At first, it was funny that Helga's family life was a mess because her sister moved out and the family just fell apart because Olga was gone (Bob became a jerk who tried to push Helga to be more like her older sister while Miriam became a covert alcoholic and just gave up being a parental figure). When Helga revealed that she's been ignored since she was pre-school-aged in favor of Olga, any humor that could be mined from it evaporated.
Especially the running gag of Bob calling Helga by her sister's name by mistake, except this time she's just a confused little girl in preschool.
Helga (tugging on her oblivious father's leg): "I have to go to preschool!"
Bob: "Huh? Oh... in a minute, Olga." (goes back to paying attention to Olga)
Helga (in a horribly pleading voice): "No, I'm Helga, dad! Helga!"
During Helga's flashback to when she first met Arnold
There's also a brief moment at the very end where Helga hugs Dr. Bliss.
Lila may be widely disliked in a lot of fan circles, but her situation in her first appearance (where she's living in a rundown apartment, her mom isn't around, and her father is unemployed and looking for a job to support his daughter) was heartwrenching.
Add to that Helga, Phobe, Rhonda, Nadine and Sheena bullying her and making her cry. Really, Lila was a flat-out Woobie in her debut.
The beginning of Arnold's Christmas where Mr. Hyunh gives up his daughter during the final days of the Vietnam War (the fall of Saigon) so she can have a better life.
The end where Mr. Hyunh actually reunites with his long-lost daughter (and the fact that Helga gave up the Nancy Spumoni boots her parents got her for Christmas just to pay the city clerk to find Mr. Hyunh's daughter and help Arnold believe in miracles again) is more tearjerking than the previous moment.
"Parents Day," where Jerkass Big Bob thoughtlessly calls Arnold "orphan-boy" at the family event - sending him into a deep depression over his parents. Later, Arnold asks Phil for the true story about his parents. Lots of fun and heartwarming moments, until the final flashback where baby Arnold says goodbye to his departing parents for what turns out to be the last time. And that dream he has after hearing the story.
"I can't find my mommy and daddy."
The situation is also pretty bad for Phil because he has to try to explain what happened to his son and daughter-in-law.
Arnold: What happened to them, Grandpa? Phil:(solemnly) They never came back; never heard from them again.
Arnold: What about their plane?
Phil: It was never found.
The scene at dinner, where Arnold, after overhearing Bob Pataki refer to him as "orphan boy", explains that he doesn't want to go back to the tournament, as it's only for children and their parents, and while he DOES love his grandparents, they aren't actually his parents.
The episode "Curly's Girl," in which Rhonda repays a debt to Curly by pretending to be his girlfriend for five days. She dumps him as soon as they agreed it would be over, and everyone hates Rhonda (even her best friend Nadine) for breaking his heart. To win back her popularity, Rhonda tells Curly to pretend to be his girlfriend again, but then HE dumps HER after hearing her in the bathroom talk about how disgusting she thinks he is (after he forcibly kissed her). As a result, people feel sorry for Rhonda and she's popular again. Then Rhonda goes to Curly's house to thank him (she thinks it was an act), and he's sitting on his front steps with his glasses off (for perhaps the only time ever in the series), eyes wide and full of tears. He didn't look like the little psychotic freak he always is— instead, he's just a depressed little boy with a shattered heart due to his unrequited love.
Any moments during which Bob acts like an actual father to Helga, such as in "Quantity Time," when he looks legitimately hurt after Helga tells him to leave her alone after he took her out for a pony ride because he thought she liked it (and thought she was "Seven... eight?" years old). Also occurred in "Arnold's Thanksgiving," when he can be seen suiting up (vest, helmet and all) like a soldier going to war because he thought Helga (and he didn't mean Olga for once) was missing, and declaring that he wasn't coming back until he found her. After Helga walked in, Bob also expressed his thankfulness for Helga as a daughter, which made Helga smile (which she rarely does in her own house). Rare as they are, these moments are probably made more tearjerker-y by the fact that Bob is usually a neglectfulJerkass to Helga.
Despite how much of a jerk he can be, it's pretty sad seeing Sid's meltdown in "Monkeyman!" after two muggers stole his boots
The part in "Chocolate Boy" where viewers find out he's obsessed with chocolate because his nanny, who always gave him chocolate, left when he was little. He cries piteously about this while hugging Arnold.
If you're a cat-lover, "Harold's Kitty" has some Tear Jerker moments, made more so because Harold was a loner and a bully at the time. The real Tear Jerker, however, occurs when he has to give his kitten back to its real owner; he comes out of the house holding it and crying, in front of the whole neighborhood. Awww.
In the Episode "Mugged", Arnold learns Karate to defend himself from muggers and to help others from being hurt. However, his incredible Karate skills go to his head, and halfway through the episode he attacks a random guy who just walked up to Arnold wanting to find directions through the town. Not only is the man now pantless, he's crying uncontrollably and slowly walks off, constantly repeating "I only wanted Directions". A small crowd looks at Arnold with complete disgust as he realizes what he's done.
In "Weighing Harold," it seems like everybody is suddenly making nasty comments about Harold's weight. His weight loss cruise fails because he was afraid the other participants would think he was fat, and ate to counteract the stress. If that's not enough of a Tear Jerker, Arnold volunteers to help Harold lose weight, but Harold initially can't do it. He breaks down crying while explaining to Arnold that everyone around him has been saying he was fat, decimating his confidence.
For anyone who has ever been bullied or used, the final scene in "Phoebe Skips," which shows Phoebe crying in the girls' room, is a Tear Jerker.