We demonstrate, by explicit construction, that a single band tight binding Hamiltonian defined on a class of deterministic fractals of the b = 3N Sierpinski type can give rise to an infinity of dispersionless, flat-band like states which can be worked out analytically using the scale invariance of the underlying lattice. The states are localized over clusters of increasing sizes, displaying the existence of a multitude of localization areas. The onset of localization can, in principle, be 'delayed' in space by an appropriate choice of the energy of the electron. A uniform magnetic field threading the elementary plaquettes of the network is shown to destroy this staggered localization and generate absolutely continuous sub-bands in the energy spectrum of these non-translationally invariant networks.