In a word, "No." Although they were related in England generations earlier, the CT and southern Brocketts came from two separate immigrants. CT Brocketts spring from John of New Haven. Southern Brocketts assuredly come from William of Norfolk Co, an immigrant of 1638. The extremely poor record-keeping in southern colonies does not allow good evidence linking generations in the south, but it is proven that William Brockett (and there was only one and he had no middle initial "E") was one of two sons of Benjamin Brockett of Craven Co, NC. Benjamin was probably son of Francis Jr, but, so far, evidence of that eludes us. The same pattern, by the way, is true of the Ives family into which William married. Both families established themselves in both CT and VA/NC. There is absolutely no reason to believe - and no evidence to support the idea - that either Brockett or Ives were directly related in America.