We develop a new strategy for studying low weight specializations of p-adic families of ordinary modular forms. In the elliptic case, we give a new proof of a result of Ghate--Vatsal which states that a Hida family contains infinitely many classical eigenforms of weight one if and only if it has complex multiplication. Our strategy is designed to explicitly avoid use of the related facts that the Galois representation attached to a classical weight one eigenform has finite image, and that classical weight one eigenforms satisfy the Ramanujan conjecture. In the case of Hilbert modular forms, under some assumptions about partially ordinary families of modular forms, we prove a similar result. If F is a totally real field in which p splits completely with v a choice of prime dividing p in F, we prove that a 1-dimensional family of v-ordinary Hilbert modular forms contains infinitely many classical eigenforms of partial weight one if and only if it has complex multiplication, conditional on a geometric construction of these families. We also relate this result to the local splitting of the Galois representation attached to a p-adic family of p-ordinary Hilbert modular forms.