It has remained unknown if the union of two linear star configurations in P2 has generic Hilbert function. In this paper, we resolve this question, proving that the union of a linear star configuration X in P2 of type s and a fat point scheme Y in P2 of multiplicity (t−1) with s≥t≥2 has generic Hilbert function. As an application, we show that an Artinian ring R/(IX+IY) of two linear star configurations X and Y of codimension 2 in P3 of type s and t with s,t≥3 has the weak Lefschetz property.