Fazbear Entertainment isn't the safest or honest company in the world. They are infamous for various incidents in which they tried to cover up from the public or even from other employees. Either these incidents are from sheer neglect or straight up malice from the company.
The name of this incident isn't confirmed to be canon, but this is one of the hidden aspects of the game lore.
The story goes that five children went missing at Freddy's during the after hours, where a man dressed in a mascot costume lured them into the back room. The suspect was found and charged, but the bodies were never found.
Following the incident, parents started to complain of the smell coming from the animatronics, as well as blood and mucus seeping from the eyes and mouths of the robots. They were described at this point to be more like "reanimated carcasses". This got the place inspected for sanitation concerns and eventually shut down by the end of the year.
The year that this incident happened is not mentioned in the game lore, but in the books. The Silver Eyes and Into the Pit says that this happened in 1985, where the former was said to happen during the summer of that year. This means that the murders happened two years after the Bite of '83 and FNaF 4.
In FNaF 1, Phone Guy says that the animatronics were originally allowed to walk around the during the day, but then was set to roam at night after one of the animatronics bit off someone's frontal lobe. This victim, according to Phone Guy is said to have survived the incident.
It was believed to be referenced again in 2, where the animatronics at that point were allowed to walk around during the day, as well as taking place in 1987.
Unlike the MCI, the Bite is shrouded in controversy in later years.
During FNaF 4's development, the teasers and the source code hints all show hints of "87" and "was it me?", implying that one of the animatronics did the Bite of '87. This implies that Scott was planning to show the Bite in the game as the final installment (at the time). However, two days after the game was released, an "83" was added in the site's source code. It took Scott three years to confirm that FNaF 4 took place in 1983.
This implies several things. Either there are two separate bites, Scott was trolling the community during 4's development, or Scott DID plan to show the Bite of '87 in that game up until it was later changes. No official comment was given, other than "1987" was never brought up in any official game since FNaF 2 (sans the FNaF World Clock Ending in the Mobile port, which jokingly said that Bouncepot did the Bite of '87). Even the Frights books never had a story involving the Bite, implying that either Scott couldn't think up of a good story or he didn't want to answer it.
Mentioned in the tapes from Phone Guy in FNaF 3. Despite the springlock being introduced and all the safety warnings that came with it, there was a incident at a "sister location" where multiple and simultaneously springlock failures occurred.
Due to this, Fazbear Entertainment deemed the springlocks to be unsafe for employees. Phone Guy mentions that replacements of the faulty springlocks will arrive, but nothing has come out of it. Strangely, the company still kept Spring Bonnie and Golden Freddy, whom are the original faulty springlocks. A bit later, it was reported that the Spring Bonnie suit was moved at the same time when Phone Guy mentioned that no customer should be taken to the Safe Room. This implies that the Missing Children Incident happened around the time of recording.
During testing of the Freddy Fazbear Virtual Experience, Jeremy points out that Glitchtrap got into the game after the hard drives from FE were scanned. He tries to warn the others, but no one listens and Jeremy was planned to be fired and discredited over what he saw.
Eventually, Jeremy's mental state gets worse until he cuts off his face with a guillotine paper slicer, presumably leading to his own death. This results in a lawsuit mentioned by Tape Girl that lead to Silver Parasol Games being shut down after development was taken over by another unnamed studio.
Unlike the other cases, this is something that Fazbear Entertainment is indirectly responsible for, yet they are also aware of this and take pride in it. As you progress in Help Wanted, HandUnit made sneering comments at Silver Parasol Games for "making a hilarious blunder", which is releasing Glitchtrap.