The reason being most people are bidialectal (able to speak two dialects), but it is usually speaking a regional dialect and the standard language. In Jennifer's case, it's a bit different; not many people are able to do that. Her "real" accent is both.
Another example is Mel Gibson, although he never speaks with an Australian accent in interviews (or as far as I know, he may not speak it when in Oz, but if he really wanted to, I'm sure he could). He was originally born in the States, moved to Oz when he was 12 (in '68) and he made his American film debut in '84. He lived an ample amount of time in both countries; that is enough to aquire both dialects. He says that he 'forced' himself to speak it, but more than likely he would've aquired it anyhow.
I have many friends who are bidialectal. One lived in England, came over to the US when he was 17. In just 5 years, he speaks with an American (Floridian) accent, but can still speak with an English (London) accent. Another raised in NYC until 9 or 10, she can speak both Queens and Floridian accent (she speaks FL around here, but produces Queens on queue). Another raised mainly in Boston and Florida, but lived around the world, so he can speak a myriad, but basically speaks either of the aforementioned.
Just a FYI: there's no such thing as a British accent (the terms English and British are interchanged too frequently in American English). It is called an English accent if the person is from England (which is what Jennifer's accent is or was). Saying British is assuming the United Kingdom, which includes England, Scotland, Wales and Northern Ireland. I assume people know each country has their own exclusive dialect and they're not indistinguishable, and that there are many, many different dialects within each country, just as there is in the US.
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