And in the US as well. There are some isolated regional accents that are virtually unintelligible to people from other parts of the country. It's not perhaps quite as pronounced as in the British Isles, but it's definitely there. But everybody understands a midwestern accent (the kind used on national news broadcasts).
For a while there was even a neutral American British accent used in major broadcasts and films...but it's since dropped by the wayside.
That's sort of confusing. In a number of places I've read an explanation that what is described in the Wikipedia article as "Mid-Atlantic English" was in fact higher class New York English before WWII (one of its most characteristic features was a non-rhotic "r"). These sources also tend to use Franklin Delano Roosevelt as an example of this sort of accent.
You're right that it's a mostly defunct upper-class accent (see Kelsey Grammer for a modern example) taught at boarding schools.
But it was also specifically taught to thespians as a preferred accent for radio, television and film performance in the early part of the 20th century. Performers like Cary Grant and Katharine Hepburn both performed in it even though it wasn't their regular accent.
In some stage acting schools it's still taught as a "neutral" accent for performance.
For a while there was even a neutral American British accent used in major broadcasts and films...but it's since dropped by the wayside.
http://en.wikipedia.org/wiki/Mid-Atlantic_English