Optical/near-infrared (optical/NIR, OIR) light from low-mass neutron star X-ray binaries (NSXBs) in outburst is traditionally thought to be thermal emission from the accretion disc. Here we present a comprehensive collection of quasi-simultaneous OIR and X-ray data from 19 low magnetic field NSXBs, including new observations of three sources: 4U 0614+09, LMC X-2 and GX 349+2. The average radio-OIR spectrum for NSXBs is α ~ + 0.2 (where Lν ~ να) at least at high luminosities when the radio jet is detected. This is comparable to, but slightly more inverted than the α ~ 0.0 found for black hole X-ray binaries. The OIR spectra and relations between OIR and X-ray fluxes are compared to those expected if the OIR emission is dominated by thermal emission from an X-ray or viscously heated disc, or synchrotron emission from the inner regions of the jets. We find that thermal emission due to X-ray reprocessing can explain all the data except at high luminosities for some NSXBs, namely, the atolls and millisecond X-ray pulsars. Optically thin synchrotron emission from the jets (with an observed OIR spectral index of αthin < 0) dominate the NIR light above and the optical above in these systems. For NSXB Z-sources, the OIR observations can be explained by X-ray reprocessing alone, although synchrotron emission may make a low-level contribution to the NIR, and could dominate the OIR in one or two cases.