The story I heard was that the accent in and around New England would be closer to Shakespeares accent than most modern British accents.
Not sure I believe it though.
As Brad says, the accent changes over small distances in the UK, and the further back in time you go, the more the accent variations existed.
If Shakespeare had an accent, chances are it was very different from those more than 10 miles away, so not that representative of the whole UK.
I was born in Stockport, and at the age of 17-18 moved to Bolton - 20 miles north - it took me 3 or 4 months to really understand the local accent well enough to stop asking people to repeat themselves - and as far as they were concerned, I was (and probably still am) a 'bloody Southerner'.