We study the evolution from suppressed to enhanced optical transmission through metal nanohole arrays with increasing film thickness. Due to Fano interferences, the plasmon resonances gradually shift from transmission dips for ultrathin films to peaks for thick films, accompanied by a Fano asymmetry parameter that increases with film thickness. For intermediate thicknesses, both peaks and dips in transmission are far from the plasmon resonances, and hence, also far from the spectral positions of maximum light absorption and nearfield enhancements. Calculations for various hole diameters and periodicities confirm the universality of our conclusions.