It puts your example of a 'fine' sample rate into context, its small. An error rate of +/-3% would require much more than a sample rate of 4,000. By pure maths alone you would be looking at around 5% of the population or in this case the turn out from the referendum to get a something around a 99% confidence level. My point back in all these posts is around trusting the opinion polls, and the fact the sample rates are so low is therefore meaning that you can take a pinch of salt with every single one. However many people are hanging their hats on these opinion polls being correct, which given the error rate of being +/-10% based on such a low sample rate, means that the gap from a 52-48 vote is so big that the opinion poll isn't worth the paper it's written on. That's just basic maths.
For someone who proclaims to be in big business (ergo "very complex business projections"), you do struggle to remain calm when you get challenged.