Two new sequential poly(ester amide)s (PEAS) derived from 1,4-butanediol, sebacic acid, and L-alanine (PABA8) or glycine (PGBG8) are prepared and characterized. For comparative purposes the related polyesters (PEs) 4,10 and 6,10 are also studied. The calorimetric analysis shows that the inclusion of amino acids improves the thermal properties such as the melting temperature without a significant reduction in their thermal stability. All polymers show hydrolytic and enzymatic degradability. The degradation rates of the PEAS are higher for the alanine derivative (PABA8) because of its low crystallinity and the higher specificity of the essayed proteolytic enzymes. The PEs are only degraded faster when enzymes with esterase activity are employed.