We examine the structure and properties of cold drawn Ti-50.1 at % Ni and Ti-50.9 at % Ni shape memory alloy wires. Wires with both compositions possess a strong <111> fiber texture in the wire drawing direction, a grain size on the order of micrometers, and a high dislocation density. The more Ni rich wires contain fine second phase precipitates, while the wires with lower Ni content are relatively free of precipitates. The wire stress-strain response depends strongly on composition through operant deformation mechanisms, and cannot be explained based solely on measured differences in the transformation temperatures. We provide fundamental connections between the material structure, deformation mechanisms, and resulting stress-strain responses. The results help clarify some inconsistencies and common misconceptions in the literature. Ramifications on materials selection and design for emerging biomedical applications of NiTi shape memory alloys are discussed.