In the present study, poly(3-hydroxybutyrate-co-3-hydroxyhexanoate) [P(3HB-co-3HHx)] was reinforced with hydroxyapatite nanoparticles (nHA) to produce novel nanocomposites for potential uses in bone reconstruction. Contents of nHA in the 2.5–20 wt % range were incorporated into P(3HB-co-3HHx) by melt compounding and the resulting pellets were shaped into parts by injection molding. The addition of nHA improved the mechanical strength and the thermomechanical resistance of the microbial copolyester parts. In particular, the addition of 20 wt % of nHA increased the tensile (Et) and flexural (Ef) moduli by approximately 64% and 61%, respectively. At the highest contents, however, the nanoparticles tended to agglomerate, and the ductility, toughness, and thermal stability of the parts also declined. The P(3HB-co-3HHx) parts filled with nHA contents of up to 10 wt % matched more closely the mechanical properties of the native bone in terms of strength and ductility when compared with metal alloys and other biopolymers used in bone tissue engineering. This fact, in combination with their biocompatibility, enables the development of nanocomposite parts to be applied as low-stress implantable devices that can promote bone reconstruction and be reabsorbed into the human body.