We show that nanoporous anodic alumina films, with pore diameters in the range 10-80 nm, can be transformed from being very hydrophilic (or super-hydrophilic) to very hydrophobic (or super-hydrophobic) by coating the surface with a thin (2-3 nm) layer of a hydrophobic polymer. This dramatic transformation happens as a result of the interplay between surface morphology and surface chemistry. The coated surfaces exhibit 'sticky' hydrophobicity as a result of ingress of water into the pores by capillary action. The wetting parameters (contact angle and contact angle hysteresis) exhibit qualitatively different dependences on pore diameters in coated and uncoated films, which are explained by invoking appropriate models for wetting.