Low dielectric loss and high electrical breakdown fields, in conjunction with an application-specific permittivity response, are simultaneous requirements for (BaxSr1-x)Ti1+yO3+z thin films. This has proved problematic, in that synthesis conditions which tend to reduce dielectric loss also tend to reduce zero-bias permittivity and electric-field tunability of the permittivity, thereby necessitating undesired compromises in overall performance for some applications. In order to address this problem of simultaneous optimization, we fabricated BST thin films with very low dielectric loss, Ti-rich BST layers ((Ba + Sr)/Ti = 0.73) at the electrode/BST interfaces, in series with a primary, high tunability, high permittivity BST layer ((Ba + Sr)/Ti = 0.9) in the center of this layered BST structure. Planar capacitors fabricated from the BST films exhibit low dielectric loss (tan delta = 0.005) simultaneously with high (76%) tunability at room temperature and high breakdown fields (similar to4 MV/cm), compared with capacitors with a single BST layer ((Ba + Sr/Ti = 0.9). Post annealing of the layered BST films at 800 degreesC further improved dielectric loss down to 0.003, while keeping the high tunability and high breakdown field of the as-deposited films.