His agent said he had been living with Parkinson's disease and died at Denville Hall, a retirement home often used by those in the entertainment industry.
A statement from the family said: "He lived courageously with Parkinson's disease for a long time."
Cant was also a guest presenter on Jackanory and appeared on ITV's Dappledown Farm, which ran from 1990 to 2003.
He got his big break when he auditioned for Play School, when part of the audition required him to climb into a cardboard box.
His voice was known to millions in shows such as Trumpton, his most famous line being the fire brigade call-out: "Pugh, Pugh, Barney McGrew, Cuthbert, Dibble, Grub."
For millions his voice immediately evokes childhood.
In the early '60s - an audition in which he was asked to sit in a cardboard box led to a job on a new programme called Play School.
Born in Ipswich, he trained as a printer before having a go at acting.
The warm, friendly voice was perfect for another children's venture - Trumpton, Camberwick Green - and the slightly more industrial Chigley.
If it sounds as though it was recorded in a cupboard, it's because it was.
Along with Play Away, Bric a Brac and other programmes he was part of children's TV for more than 20 years.
He wrote and appeared on stage, but more than anything he was - for many - a much loved part of childhood.
Cant was honoured with a special Children's Bafta award in 2010.
In a BBC interview, he explained that he had been diagnosed with Parkinson's in 1999.
He said the one thing he wanted children to take away from his work was "that I made them laugh, I made them feel happy".
On receiving his Bafta, Cant said: "One of the main rules of those Play School days was that we should play to the camera as though we were talking to one child, in whatever circumstance.
"It could be somebody in a tower block, a nice semi-detached somewhere, or a Royal palace. You had to phrase everything so, whoever was watching it, they felt you were talking to them."
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