The structure-dynamics relationships in random copolymers of poly(butylene adipate) and poly(butylene isophthalate) (PBIPBA) have been investigated by means of X-ray scattering and dielectric loss spectroscopy experiments. The results show that the-incorporation of flexible butylene adipate units into the more rigid PBI chain increases the ability to crystallize the resulting copolymer as compared to that of PBI. The subglass dynamics of the copolymers is characterized by the existence of two processes, beta' and beta" assigned to the relaxation of the individual butylene adipate (BA) and butylene isophthalate (BI) comonomeric units, respectively. The beta"-relaxation of the amorphous specimens exhibits an unexpected enhanced mobility with increasing BA molar ratio; which has been explained by considering a progressive increase of the intrachain cooperativity. The segmental dynamics for the amorphous specimens, characterized by the alpha-relaxation, follows an expected molar ratio behavior. However, an anomalous effect characterized by an enhancement of the segmental mobility induced by the crystallinity is observed in the 50/50 copolymer. This effect has been explained by considering the progressive enrichment of the amorphous phase in butylene adipate segments as crystallinity develops.