Breastfed babies are smarter because their mothers are clever in the first place, not because of any advantage of breastfeeding itself, a study suggests.
Researchers found breastfeeding mothers tended to be more intelligent, more highly educated, and likely to provide a more stimulating home environment.
However, they stressed that there were still many advantages to breastfeeding.
The British Medical Journal study was carried out by the Medical Research Council and University of Edinburgh.
Lead researcher Geoff Der said: "This question has been debated ever since a link between the two [high IQ and breastfeeding] was first discovered in 1929.
"Breastfed children do tend to score higher on intelligence tests, but they also tend to come from more advantaged backgrounds."
The researchers analysed data from more than 5,000 children and 3,000 mothers in the US.
They found that mothers who breastfed tended to be more intelligent, and when this fact was taken into account, most of the relationship between breastfeeding and the child's intelligence disappeared.
The rest was accounted for by other aspects of the family background.
The researchers also looked at families where one child was breastfed and another was not.
This confirmed the earlier results - the breastfed child was no more intelligent than his or her sibling.
Putting the results together with other studies that measured the mother's IQ confirmed this pattern.
Mr Der said: "This research shows that intelligence is determined by factors other than breastfeeding.
"But breastfeeding has many benefits for both mother and child. It's definitely the smart thing to do."
Breastfeeding has been linked to a range of health benefits.
Just one day of breastfeeding is thought to be enough to stabilise a baby's blood sugar levels, and provide natural antibodies against disease.
Breastfed babies have been shown to be less prone to diarrhoea, vomiting, and respiratory infections. Breastfeeding may also have a long impact on reducing blood pressure and obesity.
The World Health Organization recommends that babies should be breastfed for at least the first two years.
The UK has one of the lowest breastfeeding rates in Europe - almost a third of women in England and Wales never try to breastfeed, compared with just 2% in Sweden.
Rosie Dodds, of the National Childbirth Trust, said the study was not conclusive.
She said a study in the Philippines - where, unlike the West, poorer women are more likely to breastfeed - showed that breastfed children were likely to be more intelligent.
However, she added: "Women do not breastfeed because of any benefit to their baby, they do it because it feels like the natural thing to do.
"It is important that women make a decision that is right for them, and their family, and they should not be pressurised either way, but we would like to see more support for women who do decide they want to breastfeed."
The Department of Health said breastfeeding was the best form of nutrition for infants.
"We know that the composition of breast milk meets the individual needs of each baby and that, as a result, breastfeeding can make a major contribution to public health."