Through simulations and experiments we found considerable photonic bandgaps (PBGs) in two-dimensional structures with short-range periodicity (SRP) and without any spatial symmetry. A multiple-scattering method is used in our numerical investigations. Our experiments are performed in the microwave region with a microwavevector network analyser. The short-range periodic structures used in the investigations are produced through the arrangement of four certain basic units in a Fibonacci sequence in two dimensions. It is interesting that the PBG positions and widths are invariant for different translational cutouts and for different thickness of these SRP structures, showing that the PBG properties of the structures under consideration are decided by their SRP and not by their patterns. The physics behind the results is explored.