Annie's accent

Actually, you wouldn't pick up an accent from a family member that is as strong as an accent a person would have from living in a particular country. Was Annie Sullivan born in Ireland and then taken to Massachusetts? Anne Bancroft does a very good Irish accent, for someone who lived there and learned to speak there and then moved to the USA. My parents are from Ireland yet I do not have an Irish accent, nor any of the other Irish children I grew up with who were born in the USA had one but the children who were born in Ireland and who came to the USA by the first grade of school had accents that became slight but not altogether gone like Annie Sullivan's. If Ms. Bancroft was really doing a good Irish accent anyone not raised in Ireland or around Irish accents would probably have a very difficult time understanding her. As it is, Ms. Bancroft did a fine job with the accent...sometimes I couldn't understand a word here or there. That's exactly how it is when I hear Irish people speak in my everyday life.
That's how it is with me in my experience as a 1st Generation Irish-American.

~Daniel Day-Lewis doesn't break character until the night of the Academy Awards.

Both of Annie's parents, Alice and Tom Sullivan (I wonder if they were really married when they had Annie) were born in Ireland. They were very, very poor and Annie really had a hard time growing up.

Her accent is a mingle of Irish and Bostonian. She most likely picked up her parents' accents when she was learning to speak. And remember, Annie was blind, so she had to rely on her hearing much more than the sighted person.