Allport and Cattell believed we were all born with five core personality traits

Allport and Cattell believed we were all born with five core personality traits:  openness, conscientiousness, agreeableness, extroversion, and neuroticism.  If these five, as you pointed out above, neuroticism is most closely associated with anxiety.  Therefore, they believed that if one was born into a family where neuroticism was high, chances are the person could have higher levels of anxiety verses those who were born in families where neuroticism was low.  Do you agree?