It was something of the theme of Sybill Trelawney's arc and something that has been discussed at length in the HP fandom for years. Trelawney, while in many ways a total fraud, actually does make many accurate predictions but is cursed not to be believed and/or they are dismissed because they don't pan out as expected.
Ironically her two "true" predictions (The Harry/Voldemort "Chosen one" prophecy and the Wormtail "Servant rejoins his Master" prophecy) are the ones she is unaware of making, whereas all her throwaway, inconsequential ones that don't pan out as expected are the ones that people listen to and know about.
There are reasons for this though - the main one is the link to the doomed seer Cassandra in Greek Mythology. In OoTP, it's said that Trelawney is the descendant of the great seer Cassandra Trelawney. While it might seem like a throwaway line, it probably refers to the Greek prophet Cassandra (JKR was a huge fan of Classics). In the Greek myth, Apollo fell in love with a mortal Cassandra and gave her the gift of prophecy. He then tried to put the moves on her and she rebuked him. He got angry and instead of taking the gift back, he cursed her: nobody would believe her predictions or the predictions of her descendants.
Also there's the etymology of her name - "Sibyl" was a priestess in ancient Roman mythology. A sibyl is a woman who could look into the future and the name is from the Latin sibylla, seer. There is also the reference to the "Sybil" from the Sybilline Books, a Roman collection of oracular occurrences.
A potted history of Trelawney's minor predictions and how they were fulfilled (mostly in an unexpected way):
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She told Parvati Patil to beware of a red-headed man. Parvati immediately suspected Ron. At the Yule Ball, Harry and Ron, Parvati's and her sister, Padma's dates, ignored her and her sister. Two years later, Ron started going out with Parvati's best friend Lavender Brown, who started to ignore Parvati.
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She predicted that the thing Lavender was dreading would happen on 16 October. On that day, Lavender got a letter that her pet rabbit, Binky, had been killed by a fox.
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In the beginning of the 1993–1994 school year, she told in the whole class that, near Easter, one of their number will leave them forever. Everyone assumed it was a reference to Harry someone dying but actually Hermione quit the class in disgust near Easter holidays.
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In GoF Trelawney, while discussing star charts, commented that given his "dark hair, mean stature, [and] tragic losses so young in life..." Saturn was likely in a position of power when he was born, and suggested that he was born in mid-winter. Interestingly (and unknown to both at that time), whilst Harry was born in summer (July 31st) and therefore her prediction was wrong for him Harry was a Horcrux of Voldemort, who was born on New Years Eve and also suffered loss with his mother's death just after his birth.
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At the Christmas lunch, Trelawney at first refused to join the table, since when thirteen dine together, the first to rise will be the first to die. Harry and Ron got up from the table first together, but Dumbledore was the first among those at the table to die. Most would assume that was her prediction being wrong ... but she did not take into account Peter Pettigrew (at the time disguised as Scabbers). Before she was seated, there were thirteen people at the table (counting Scabbers) and when Dumbledore rose to greet her he was the first person at the table of thirteen to rise, and was the first to die.
This "prophecy also happens at other times when 13 people are dining together in the books and whoever rises first is the next one to die of the group, but none are directly foretold by Trelawney in the moment: In 1995, thirteen members of the order dined together, and the first to rise was Sirius, who was the first to die of those thirteen people ... Also, after the Battle of the Seven Potters, thirteen people (Mr and Mrs Weasley, Bill, Fleur, Fred, George, Ron, Ginny, Harry, Hermione, Lupin, Tonks and Hagrid) gathered in The Burrow to mourn the death of Alastor Moody. Afterwards, Lupin stood up and offered to look for Moody's dead body. Later in the Battle of Hogwarts, Lupin was killed by Antonin Dolohov. Fred and Tonks were also killed, but as the time of Lupin's death is unknown, it is possible he was first.
Unrelated to Trelawney but since I just brought up the Marauders - one other little "easter egg is that the four Marauders - Moony, Wormtail, Padfoot and Prongs - die in reverse order of their names on the map - James, Sirius, Peter, Lupin
- In HBP Trelawney meets Harry in the corridors, and complains that Dumbledore has been repeatedly dismissing her warnings of impending disaster. She looked at some Tarot cards and saw, "... the lightning-struck tower... Calamity. Disaster. Coming nearer all the time...". This seems to predict the events of Dumbledore's death on the Astronomy Tower.
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In OoTP Umbridge insists that Trelawney demonstrate her abilities by prophesying something. Trelawney claimed that she saw dark events ahead for Umbridge, which Umbridge immediately dismissed as nonsense. Later, Umbridge was dragged off and imprisoned by centaurs in the Forbidden Forest.