Is the key to clean washing results, Rinsing?

I don't know which manufacturer it was in the old days, more than likely Hotpoint, but they used to have listed in the instruction book and in their leaflets and other media (I think even mentioned in TV adverts) that the secret to clean washing was not so much the washing cycle but that it was the rinse cycle that got the clothes clean so they boasted that there were 3 rinse cycles to get your clothes extra clean.

So is that a true claim do you reckon? is the secret in the rinsing? - most modern washing machines these days have only 2 rinses, especially on a quick wash (1 rinse and 1 rinse with fabric conditioner)