Volumetric left-handed metamaterials made up of an array of split-ring resonators (SRRs) and wires exhibit negative index of refraction in a very narrow bandwidth due to the resonant nature of SRRs. We investigate the possible bandwidth enhancement by adding resonances to the system using fractals. The operating bandwidth of the system is increased when the additional resonances are placed close enough to each other. The Sierpinski-carpet fractal pattern is chosen as the distribution for the SRRs. The principle is demonstrated through simulations, and prototypes are fabricated and tested to verify consistency with simulations.