In one embodiment, the present invention relates to a non-woven fiber assembly comprising one or more fibers wherein each fiber contains: a hydrophilic component; and an elastomeric component, and wherein the non-woven fiber assembly further comprises an adhesive component. In still another embodiment, the present invention relates to a non-woven fiber assembly comprising one or more fibers wherein each fiber contains: a hydrophilic component; an elastomeric component; and an adhesive component, wherein the hydrophilic component, the elastomeric component and the adhesive component are all contained within each fiber. Also disclosed is a method of making the afore-mentioned non-woven fiber assemblies. Additionally, a medical dressing made from the non-woven fiber assemblies of the present invention is disclosed.