How I Met Your Mother, like most sitcoms of this style, is about friends who are so close that they are practically family - and always do everything together. Despite this, the core gang has their fair share of falling outs over the seasons, along with breakups (both within the MacLaren's crew and with other people) and a few other forms of separation along the way.
Some of these are minor, or don't become a large part of the story. When Robin took a step back from the gang to focus on her relationship with Don, it didn't feel like a huge loss for the gang. In other cases, though, the fans' hearts broke when these pairs broke up.
When Ted Was Left At The Altar
The closest that Ted gets to marriage before meeting the mother is when he and Stella are engaged, and they get all the way to their wedding day on Shelter Island. At the last moment, though, Stella leaves him at the altar, running off with her ex (in her wedding dress!) and leaving behind a note. This would be gut-wrenching for anyone, let alone someone as obsessed with finding "the one" as Ted is. It also impacted the character (and the show) for a long, sad, time to come.
When The Missed Robots Vs Wrestlers
'Time Travelers' is a particularly unusual episode of the show, as Ted decides whether or not to go to Robots Vs Wrestlers, assisted (or hindered) by various future versions of himself and Barney (while Robin and Marshall dance-off over a drink). In the end, though, the episode reveals that Ted was alone at the time, and that all his friends were too busy to come. Given their history (as a group) with Robots Vs Wrestlers, seeing this scene as they grow up and move on was crushing.
When Robin & Barney Divorced
Robin and Barney became a fan-favorite couple, but in the finale flash-forwards, it is revealed that they divorce. The two are seen fighting while traveling for Robin's job, and it becomes clear that they are always fighting, and deeply unhappy as a couple.
A later flashback sees them telling their friends that they decided to get divorced. After seasons of build up, so much attention paid to their wedding, and such an incredible history, seeing them end things because they just couldn't make it work was heartbreaking.
When Barney Was Abandoned By His Father
Barney's lack of a father-figure is a recurrent theme in the show, and is covered in flashbacks, scenes with his mother, and of course, scenes when he later meets his biological father and learns that he became a 'boring' family man in the suburbs. Seeing how growing up without a Dad impacted Barney is sad enough, but HIMYM really drove it home with a scene when Barney goes to have dinner with his father at home - and ends up storming out, and trying to steal the basketball hoop. In a painfully emotional scene, he yells at his father that if he was 'going to be some lame suburban dad, why couldn't you have been that for me?'.
When Marshall's Dad Died
Probably the most famously heartbreaking episode in the series is when Marshall's father unexpectedly dies - but there is more depth to this separation than that. Marshall often struggles with being so far from his family, and this episode deals with him being unaware of how to talk to his father about trying to get pregnant, and fears that there may be a medical issue. It's layered and nuanced, incredibly well put-together, and deeply emotional to see this come together with the news of his ing.
When Lily & Marshall Left Marvin At Home For The First Time
This was only a temporary separation, but one that resonates with any new parents. After Lily and Marshall had baby Marvin, the first time that they left him with someone else for the night, it didn't go quite as they expected.
They thought they would love being out on their own, but instead, found themselves dressed up at Ted's event, sniffling over how much they missed their baby. It was funny and cute, but a little heartbreaking too.
When Barney & Ted Stopped Speaking
Probably the Barney and Robin slept together (after Robin and Ted broke up), and Ted was so furious with Barney that they stopped speaking entirely. For a while, it looked like this might be it for the two bros, and it was only a major traffic accident that allowed them to rebuild the friendship. Much as Ted and Barney are far from perfect, they were incredible friends, and it was tough to see them go through this.
When Tracy Died
Although Tracy (the mother) only appeared in the final episodes of the show, and in the flash-forwards, fans were invested in her - and when she became ill and died at a tragically young age, fans were heartbroken. Flash-forwards saw her being diagnosed, and with Ted at her side in a hospital bed. This was such a painful end, in fact, and fans were so vocal about their anger over it, that the HIMYM DVD box set includes an alternate ending where she doesn't die.
When Lily Left Marshall For San Francisco
All the way back in the first season, Lily is struggling with her dreams of being an artist, and her fears that she will never realize them now that she is an engaged teacher in New York. At the end of the season, she leaves Marshall, breaking off their engagement and heading to an art program in San Francisco, and fans see Marshall sitting on the stoop, in the rain, holding the ring. The separation itself was heartbreaking, but so was the fallout: watching Ted get angry at Lily for cutting him off, too, watching Lily struggle to come home and make things work again, and seeing Marshall struggle to get over it.