We present low and high resolution long-slit spectra of three bipolar planetary nebulae (PNe) with bright central cores: A 79, He 2-428 and M 1-91. He 2-428 and M 1-91 have high density (from 103.3 to 106.5 cm-3) unresolved nebular cores that indicate that strong mass loss/exchange phenomena are occurring close to their central stars. An F0 star is found at the centre of symmetry of A 79; its reddening and distance are consistent with the association of the star with the nebula. The spectrum of the core of He 2-428 shows indications of the presence of a hot star with red excess emission, probably arising in a late-type companion. A 79 is one of the richest PNe in element{N} and element{He}, the abundances of M 1-91 are at the lower end of the range spanned by bipolar PNe, and He 2-428 shows very low abundances, similar to those measured for halo PNe. The extended nebulae of A 79 and He 2-428 have inclined equatorial rings expanding at a velocity of ~ 15 km s-1, with kinematical ages >=104 yrs. The association of these aged, extended nebulae with a dense nebular core (He 2-428) or a relatively late type star (A 79) is interpreted as evidence for the binarity of their nuclei. Based on observations made with the 2.5 m INT and the 2.6 m NOT telescopes operated on the island of La Palma by the ING and the NOTSA, respectively, in the Spanish Observatorio del Roque de Los Muchachos (ORM) of the Instituto de Astrofísica de Canarias.