The present work focused on the development and characterization of biocomposites based on a fully bio-based polyester, poly(butylene succinate-co-butylene adipate) (PBSA), and wheat bran derived by flour milling. PBSA-bran composites containing 5, 10, 15, and 20 wt.% of wheat bran were produced via melt extrusion and processed by injection molding. Their thermal, rheological, morphological, and tensile properties were investigated. In addition, a biodegradation test in a natural marine environment was conducted on composite dog-bones to assess the capacity of the used filler to increase the PBSA biodegradation rate. The composites maintained similar melt processability and mechanical properties to virgin PBSA with up to 15 wt.% bran content. This result was also supported by morphological investigation, which showed good filler dispersion within the polymer matrix at low-mid bran content, whereas poor polymer-filler dispersion occurred at higher concentrations. Furthermore, the biodegradation tests showed bran’s capacity to improve the PBSA biodegradation rate, probably due to the hygroscopic bran swelling, which induced the fragmentation of the dog-bone with a consequent increase in the polymeric matrix–seawater interfacial area, accelerating the degradation mechanisms. These results encourage the use of wheat bran, an abundant and low-cost agri-food by-product, as a filler in PBSA-based composites to develop products with good processability, mechanical properties, and controlled biodegradability in marine environments.