What is the meaning of the idiom "cat's in the cradle"?

I heard the expression "cat's in the cradle" for the first time in the song by the band Ugly Kid Joe, I though at first that it was just something they came up with and I did not think there was any common meaning to that. But, couple of days ago I heard this expression again in a TV series. Does anyone know what this means?

I may have been missing something here, but my interpretation was that the chorus (of the song Cat's in the Cradle) was referencing childrens stories. Presumably ones that the father in the story never had time to read to the child.

The silver spoon is assumed to be the one the dish ran away with.

Little Boy Blue was Published in L. Frank Baum's first children's book, Mother Goose in Prose (1897)

The man in the moon is a nursery rhyme

The line 'The Cat's in the Cradle was, to my mind, a reference to 'The Cats and the Cradle', a Dutch Fairy tale

Without knowing the context of how it was used in the tv show, it's hard to know whether they were simply referring to the song, or if they intended meaning from the Fairy Tale the song is (I believe) referring to.

snopes.com discusses an (obviously untrue) urban myth about cats smothering new born babies, so cat is in the cradle may be a reference to that old wives tale, with the implication that a cat in the cradle is dangerous and implies the baby is forgotten and neglected.

CLAIM: Cats suck the breath from babies, sometimes killing them. FALSE.
The idea that a cat could suck the breath of an infant is simply a misguided notion — cats just don’t do that. It is said the smell of milk on the child’s breath draws the feline in for the kill, but anyone who has been around housecats knows the average moggie doesn’t much care for the liquid. (Given free choice between plain water and a bowl of milk, cats generally head for the water unless milk has been the only liquid offered to them from weaning onwards. Put more simply, unless the cat has been taught to like milk, it generally won’t seek out that substance on its own.)

Another theory advanced as to why a cat would want to harm a baby relates to the jealousy the pet will supposedly experience when the little bundle from heaven is brought into the household. No longer the center of attention, the neglected pet is allegedly capable of setting about to get rid of what it sees as the usurper. This theory is of far more recent coinage than the bit of lore it purports to explain, though, coming into fashion no earlier than the 20th century (while the “smother” belief dates to at least the 1700s).

In 1791 a jury at a coroner’s inquest in England rendered a verdict to the effect that a Plymouth child had met his death by a cat sucking out its breath. The superstition itself is older, with print sightings of it recorded from 1607 and 1708, so that 1791 verdict should be viewed with the realization that the jury was probably influenced by a snippet of “everybody knows” lore when it came time to explain a death for which there was no apparent cause.

Note: this answer is now completely rewritten after Jason C noticed problems with the Wikipedia article that I had originally referenced.