Could a star become a planet? I am asking this because the gas giants are ¨Failed Stars¨ and they are classified as planets in our solar system.
A star could become a planet, but this almost never happens. It would probably have to lose almost all of its mass early in its life, perhaps by a black hole. Then, the exposed core would have to be stopped from becoming degenerate and further mass loss would have to continue. It is possible, but it is not realistic and will not happen in a human lifetime, even if you observed every star in the universe (including white dwarfs).
A brown dwarf definitely can become a planet. A brown dwarf early in its life acts like a star, fusing deuterium and reaching temperatures of almost 2000 K. But as it ages it cools down and stops fusing deuterium. Then it acts like a planet.
In theory, a brown dwarf directly could become a planetary-mass object through mass loss. This could happen from a giant impact or another body accreting from it.
Contrarily, a planet (brown dwarf) can become a star if it accretes enough mass. For example, the most massive brown dwarfs have masses of about ~80 Jupiters, while the least massive stars have a mass of about ~60 Jupiters (there is an overlap). Therefore, a "planet" could become a star under special circumstances.
As @User123 said in a comment, all stars start as a "planet," without burning hydrogen. While it accretes more material, fusion ignites in its core and it becomes a star.