In this work we propose and experimentally demonstrate a novel non-orthogonal multi-band carrier-less amplitude and phase (NM-CAP) scheme for bandlimited visible light communication systems in order to increase the spectral efficiency. We show that a bandwidth saving up to 30% can be achieved thus resulting in 44% improvement in the measured spectral efficiency with no further bit error rate performance degradation compared to the traditional m-CAP scheme. We also show that higher order systems can provide higher bandwidth compression than low order systems. Furthermore, with no additional functional blocks at the transmitter or the receiver the proposed scheme introduces no extra computational complexity.