WMG / I Dream of Jeannie

The reason Jeannie's entire family are magical when she frequently mentions growing up as a human is because the curse put on her when she refused to marry the blue djinn extended to her entire family. Jeannie 2 was once a normal human girl who just wanted to dance and have a good time. Jeannie refused to marry and in return, gets a two thousand year nap and a handsome, loving (if not uptight) master who instantly frees her and doesn't demand anything from her. Jeannie's sister got two thousand years of involuntary servitude under a series of cruel, stupid and neglectful masters. She takes full advantage of having magical powers but that doesn't mean she asked for them.
  • Considering Evil Jeannie had hundreds of past lovers in one episode, it is also possible that she was, well, naughty and as turned into a genie as punishment.

Celebrity Paradox applies to both I Dream of Jeannie and Bewitched

Each is a television show in the other show's universe.

The Animated Adaptation Jeannie is another sister of live-action Jeannie
Both are favorites of Great Hadji; both were given difficult apprentices to chaperone; both have a similar attitude about their 'masters'. Animated Jeannie was imprisoned while searching for her sister, and washed up in the California surf, found by Corey.

Jeannie was turned human as a baby
As a variant on the above WMG about why Jeannie mentioned growing up human but was then established to be from a family of genies, perhaps Jeannie's family were genies originally but she was made human as a baby either as a reward for her family's service (so she could live a free life) or the Djinn did it so she could grow up to be a proper wife to him rather than just a servant (given Djinn are immortal it's not unreasonable to think he may have planned a marriage a couple of decades in advance). Then when she refused to marry him she was turned back into a genie as punishment.