Fifteen years after the first experimental evidence of increased oil recovery by low-salinity water injection (LSWI), clear understanding of the mechanisms has not emerged yet out of more than 500 published laboratory experiments. Firstly, it is shown that there is increasing experimental evidence that published tertiary LSWI corefloods do not often succeed in increasing significantly the recovery within the two to three first pore volumes (PV) of tertiary injection, despite strong claims of positive results. This paper focuses on sandstones and primarily on studies where secondary LSWI performs better than secondary high-salinity water injection (HSWI). Even in such cases, some examples show that the efficiency of tertiary LSWI may range from poor to nil. These cases satisfy all "required conditions," such as presence of clay, of connate water, and mixed wettability. Conditions of existence of a double saturation shock, effects of dispersion in the water phase at S-orwHS, and the direction of wettability modification are the hypotheses discussed in this paper to understand the poor performance of tertiary LSWI. Some key experimental observations are then compared to these possible explanations. They may explain the vast majority of published studies, but counter examples can also be produced for any single proposed mechanism. This paper also provides evidence that some types of experimental measurements have been neglected and deserve more attention, comments on the effect of interfacial tension, and suggests a new approach for investigating the efficiency of tertiary low salinity water flooding.