In this paper, we experimentally demonstrate the performance of a non-orthogonal multi-band super-Nyquist carrier-less amplitude and phase (m-SCAP) modulation for visible light communications (VLC). We break the orthogonality between sub-bands in the frequency domain by compressing the spectrum, purposely overlapping them, and introducing inter-band interference (IBI). We demonstrate that our proposed system can tolerate IBI, and hence spectral efficiency can be increased without introducing additional complexity to the receiver. We show that m-SCAP can tolerate up to 30% and 20% compression for 4- and 16-level quadrature amplitude modulation, respectively, thus leading to an improvement in spectral efficiencies up to 40% and 25%, respectively, at the cost of bit error rate performance, which however remains below the 7% forward error correction limit. Moreover, the experimental results are supported by numerical simulations.