Nanotwinned metals are attractive in many applications because they simultaneously demonstrate high strength and high ductility, characteristics that are usually thought to be mutually exclusive. However, most nanotwinned metals are produced in polycrystalline forms and therefore contain randomly oriented twin and grain boundaries making it difficult to determine the origins of their useful mechanical properties. Here, we report the fabrication of arrays of vertically aligned copper nanopillars that contain a very high density of periodic twin boundaries and no grain boundaries or other microstructural features. We use tension experiments, transmission electron microscopy and atomistic simulations to investigate the influence of diameter, twin-boundary spacing and twin-boundary orientation on the mechanical responses of individual nanopillars. We observe a brittle-to-ductile transition in samples with orthogonally oriented twin boundaries as the twin-boundary spacing decreases below a critical value (similar to 3-4 nm for copper). We also find that nanopillars with slanted twin boundaries deform via shear offsets and significant detwinning. The ability to decouple nanotwins from other microstructural features should lead to an improved understanding of the mechanical properties of nanotwinned metals.