The American academics noted their results were "conservative" compared with earlier studies of fewer children, which found up to 10-fold increases in behavioural problems for those with breathing difficulties in sleep.
However, they said their report, published in the US journal Pediatrics, used a much larger sample which followed children over a long period, rather than just being a 'snapshot' in time.
Karen Bonuck, an epidemiologist at the Albert Einstein College of Medicine in New York, and lead author of the study, said their approach of following children over time provided some evidence that sleep breathing problems actually caused behavioural problems.
She said there were "lots of potential reasons" why poor sleep breathing could cause behavioural problems later in childhood.
"Just think about the brain's rapid development in the early years, the need to reset itself at night, and the lack of oxygen and excess carbon dioxide that quite possibly could result from sleep-disordered breathing," she explained.
There was evidence that the pre-frontal cortex, heavily involved in controlling emotional behaviour, was particularly affected, she added, although she emphasised she was not a clinician.
She also cautioned: "It is hard to establish causality and certainly this study doesn't do that."
Other factors could come into play, the most obvious being whether children from less advantaged homes are worse behaved at four and seven years.
However, Dr Bonuck said that when they compared the apparent effects of family background on behaviour, they were far less influential than whether a child had sleep breathing problems.
They also controlled for such factors when looking at the link between sleep breathing and behaviour.