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A thought struck me while reading these questions.

Suppose there was a magical child in a pureblood family, who shows signs of being magical early and gets their Hogwarts letter on their eleventh birthday. They arrive at Hogwarts and start their education, but shortly afterwards, lose their ability to do magic. Not that they are a bad witch or wizard but they are completely incapable of doing magic. Is it possible?

Conversely, is it possible for a muggle to gain wizarding powers through teenage years, having no magical family?

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Let's address the second question first, because it's the easiest to answer. JKR has addressed this herself in an interview:

Will there be, or have there been, any "late blooming" students in the school who come into their magic potential as adults, rather than as children? ...

JKR: ... No, is the answer. In my books, magic almost always shows itself in a person before age 11; however, there is a character who does manage in desperate circumstances to do magic quite late in life, but that is very rare in the world I am writing about.
Barnes and Noble interview, March 19, 1999

While she states that "there is a character who does manage in desperate circumstances to do magic quite late in life", that never actually happens in the books. It appears that's a plot point for a later book that was eventually scrapped. There's a bit of additional information on that in this question.

As for the first question, it's possible for magical ability to be affected to varying degrees by emotional trauma. There are three examples of this in canon:

  • Nymphadora Tonks. She (temporarily) loses her powers as a Metamorphmagus in the sixth book, after being rejected by Lupin. Since the two were married, and had a child, by the end of the seventh book, she likely would have regained them.

  • Ariana Dumbledore. She lost the ability to control her magic after being attacked by a group of Muggle children, though she did still retain magical ability.

  • Merope Gaunt (Voldemort's mother). She spent most of her life being tormented by her father and brother, who believed she was essentially a Squib. However, Dumbledore speculates during his lessons with Harry in his sixth year that her magical talents would have flourished when they were both sent to Azkaban. She would later lose them again after being rejected by Tom Riddle, Sr. while pregnant with their son.

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    "however, there is a character who does manage in desperate circumstances to do magic quite late in life, but that is very rare in the world I am writing about."... so my question is, is the character JKR is referring to in that quote not Merope Gaunt? Seems to fit the description mentioned in your answer. – The Giant of Lannister Jun 27 '14 at 17:19
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Permanently losing magic doesn’t seem to be possible.

We know of wizards who were bad at magic (like Neville and Ron in their first years), and wizards whose magical abilities were weakened due to emotional issues (like Merope Gaunt, Ariana Dumbledore, and to a lesser extent Tonks - she seemed to lose her Metamorphmagus ability but retained the ability to cast spells). However, there’s no known examples of anyone who permanently lost their magic power with no way of getting it back. If someone shows early signs of magic, and then is accepted to Hogwarts, that means they are a wizard and will always be one.

It’s not possible for someone to gain magic if they don’t have it.

Albus Dumbledore, who would have no problem with the idea of a Muggle becoming magical if it was possible, refers to the king in “Babbitty Rabbity and Her Cackling Stump” as foolish for thinking he can simply become a wizard.

“The King in Beedle’s story is a foolish Muggle who both covets and fears magic. He believes that he can become a wizard simply by learning incantations and waving a wand.17

He further explains in his notes that wizards are born with their abilities, they can’t be created, and Muggles will not be able to do magic. It is possible to get magical powers without any known ancestors being wizards, though it’s likely that there would be one somewhere in the line even if they’re many generations removed.

17 As intensive studies in the Department of Mysteries demonstrated as far back as 1672, wizards and witches are born, not created. While the “rogue” ability to perform magic sometimes appears in those of apparent non-magical descent (though several later studies have suggested that there will have been a witch or wizard somewhere on the family tree), Muggles cannot perform magic. The best — or worst — they could hope for are random and uncontrollable effects generated by a genuine magical wand, which, as an instrument through which magic is supposed to be channeled, sometimes holds residual power, which it may discharge at odd moments — sec also the notes on wandlore for “The Tale of the Three Brothers.”
- The Tales of Beedle the Bard

Whether a child has magical powers seems to always be detectable by eleven. However, it’s possible that the child themselves might not know that they are a wizard, or that they might be kept out of wizarding society. (Credence Barebone in the Fantastic Beasts movie is a likely example of this - he’s untrained in magic despite his age, and it’s not clear whether anyone tried to get him into Ilvermorny or any other magic school.)

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