I don't know to how many generations back it applies but certainly to some extent if you can trace that you are descended from an Irish citizen you may be able to claim a Republic of Ireland passport and then have dual British/Irish nationality.
And it certainly seems that if you were born in a certain part of the UK, namely Northern Ireland, that not only can you presently claim dual nationality but that it is planned that you should be able to continue to do so after Brexit.
Did someone say something about the luck of the Irish?