In order for electronic textiles to truly qualify as textiles, they must maintain one of the intrinsic qualities of textiles, flexibility, or the ability to resist permanent deformation under bending, lateral stress and strain. Flexibility will allow electric textiles to be intimate, soft, wearable, conformable and durable. Unfortunately, flexibility is poorly understood by many researchers who come from a traditional electronics background. This paper presents some common terminology of textiles, and different approaches to understanding flexibility in fibers and yarns. Because one of the most mechanically stressful textile manufacturing process is machine sewing and embroidery, this paper defines the necessary properties of machine sewable yarns and demonstrates a formal Curl Test for judging the sewability and flexibility of stainless steel yarns. This paper also examines flexibility in yarns and fibers, historically and based on a mathematical model and more qualitative properties.