Electrospinning allows the production of fibrous networks for tissue engineering, drug delivery, and wound healing in health care. It enables the production of constructs with large surface area and a fibrous morphology that closely resembles the extracellular matrix of many tissues. A fibrous structure not only promotes cell attachment and tissue formation but could also lead to very interesting mechanical properties. Poly(3-hydroxybutyrate-co-4-hydroxybutyrate) (P(3HB-co-4HB)) is a biodegradable polyester that exhibits a large (>400%) elongation before failure. In this study, synchrotron X-ray phase contrast imaging was performed during tensile deformation to failure on a non-woven fiber mat of P(3HB-co-4HB) fibers. Significant reorientation of the fibers in the straining direction was observed, followed by localized necking and eventual failure. From an original average fiber diameter of 4.3 µm, a bimodal distribution of fiber diameter (modal diameters of 1.9 and 3.7 µm) formed after tensile deformation. Extensive localized necking (thinning) of fibers between (thicker) fiber–fiber contacts was found to be the cause for non-uniform thinning of the fibers, a phenomenon that is expected but has not been observed in 3D previously. The data presented here have implications not only in tissue regeneration but for fibrous materials in general.