It's the expensive private schools they went to.
Contrary to popular opinion, John F. Kennedy didn't have a 'Boston' accent. No one in Boston talks like that unless they've been to a posh prep school.

I think the descriptor for this is Mid-Atlantic accent, something common among upper class folks, especially those who attneded the "right" schools. I can't see tht the Kennedy's tlked that way, entirely.

Is it a cheap shot if it's true?

The "lower class" and then you name ethnicities. Cheap shot.

No upper-class New Yorker had a NYC/NJ accent. Only the lower classes have the traditional NYC accent, particularly people who are first or second generation Americans from Italy, Puerto Rico, or who are Jewish. Remember that many upper-class New Yorkers would spend their summers in New England and would often go to private boarding schools outside of New York.
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If anyone remembers Governor Tom Kean from New Jersey or George Plimpton the writer, they had the metro New York upper class accent. That's what Jane had, and it was utterly charming coming from her.

She was raised an East Coast cake eater. <<
Oh, that's hilarious ! And sounds truly APT !
Flanagan

This thread shows people who post are not from Upper Hudson Valley nor New England. When Jane Wyatt grew up, the northeastern part of the United States had accents that carried a British flavor to them. If you were a wealthy New Englander, your accent was very posh, British-sounding. Listen to the Roosevelts and the Kennnedys on film. That was a common accent of the time. As television has taken over, it has homogenized a lot of accents. Many people moving from one part of the U. S. to the other has further weakend regional accents. Check out Kelsey Grammer playing in "Frazier". His character spoke posh New Englander because he was well-educated and moved in wealthy circles.

She was raised an East Coast cake eater.
This will be the high point of my day; it's all downhill from here.

Having watched way too much classic TV, I'd say the worst offenders (no offense to these great ladies) are Jane Wyatt, Arlene Francis, and the woman who played Mrs. George Wilson on Dennis<<
Arlene Francis and the woman who played Mrs. George Wilson had a very pronounced accomplished, New York accent.
Jane Wyatt had a very pronounced and different sort of accent. I can't figure hers out except that actress Gene Tierney has the same, unfortunately low-class sounding accent. I always felt like Tierney sounded like Bugs Bunny. I guess they just couldn't get that hard vowel sound to make itself known.
My Dad was from New Jersey and he and his friends all "talked the same" and it wasn't like Jane Wyatt
Flanagan

It wasn't affected for Arlene Francis or Jane Wyatt. Certainly Natalie Schaefer however put it on. I can't remember Mrs. Wilson well enough.

I've always been curious about that 'upper class', affected, 'actress-y' accent. Is there a name for that? Theater becomes 'The-a-tah', father becomes 'fath-ah', car becomes 'cah', and, most annoyingly, dear becomes 'dee-ah'. They seem to have something against the letter R. And not in the same way as Brooklynites or Bostonians. Sort of like a faux British accent, as if that's something to aspire to.
Having watched way too much classic TV, I'd say the worst offenders (no offense to these great ladies) are Jane Wyatt, Arlene Francis, and the woman who played Mrs. George Wilson on Dennis The Menace. (Thank you Antenna TV). Honorable mention to Mrs. Milburn Drysdale on The Beverly Hillbillies, who spoke with an exaggerated Boston accent to great comic effect.

Not everyone from New York or New Jersey has an accent associated with that area! One assumes that her accent was naturally developed from the way her family spoke - remember we're speaking of a very uppper class gal!

Good question. I had always assumed she was British before reading her bio here.