However, when cameras were placed in the boy's room overnight, they discovered Billy was being kept awake by the families pets.
The dog was caught on camera stealing his pillow, while the cat made itself comfortable on the bed.
Robyn admitted on the show that she sometimes struggled to control Billy and his tantrums, and had tried reading to him for three hours a night to get him to sleep.
She said on the show: "Billy will finally give in about 11, 12 o'clock but he will just keep getting up God knows how many times in the night."
Robyn suspected autism because Billy's brother also suffers from the condition.
In the episode, GP Dawn Harper and Dr Ravi Jayaram observe children with behavioural problems.
Dr Jayaram didn't believe Billy had autism. He said: "His imagination was just tremendous and that doesn't really tie in with typical autism. They're very literal and that's that."
CCTV cameras were installed in Billy's room to investigate his sleep patterns and Debs Sugden, a registered nurse and sleep expert, observed the footage.
Robyn said she had thought that Billy was unable to sleep because his brain was "constantly going", but that wasn't the reason at all.
Sugden showed the mother the footage, which revealed the family dog stealing his pillow, and the cat pushing him out of bed.
This is what happened:
Sugden said: "[The dog] is on Billy’s pillow. So Billy can't even put his head down.
"So the dog has actually pushed Billy out of bed. So you're popping him back and he's going back in bed absolutely beautifully ... and then, who joins Billy in bed? The cat! He's literally shoving him out."
Sugden explained that broken sleep and a delay in falling asleep can cause concentration and behavioural problems. She said sleep was "key" for small children.
At the end of the show and after changing the pet's sleeping arrangements, Robyn said Billy was "a lot happier" and his sleep patterns improved.
She added: "We don't have so many outbursts these days.'
Born Naughty is on Channel 4 Thursdays, 8pm