It is hard to envisage how it might happen but it does. Both toddlers, domestic cats and puppies are in danger of being strangled by the cords on blinds. The recommendation nowadays is that home owners with companion animals should buy cordless blinds or ensure that in some way the cords are tied back to render them safe.
One supplier of blinds advises that “there is a possibility that over time they [cords] will intertwine and twist therefore forming a loop. A child or pet may put their head through the intertwined loop and strangle”.

If you don’t want to buy cordless blinds then you are advised to replace them with new child friendly blinds or completely remove the cords. An alternative would be to tie up the cords on blinds using cord tidies, cleats or special clips. With respect to kids, keep their cots, beds, playpens and high chairs away from windows.
In the UK (believed), there is the Window Covering Safety Council, an organisation dedicated to educating caregivers and parents about the potential for strangulation by cords and window blinds and other window coverings.
There’s been a catalogue of deaths from strangulation. Records from the Consumer Product Safety Commission in the USA indicate that cords are a “particularly insidious hazard” and they cite 41 deaths human between 1973 and 1980. In 2008, 18 children were fatally strangled by cords.

There’s no doubt that to those in the know, cords hanging from blinds are dangerous to pets and children.
On the Reddit.com website there are stories of cats being hung by blinds. In one oxample, a contributor commented, “My friend’s cat unfortunately got stuck in the cords and hung itself. They are not pet friendly. You can buy a special clip to put on the wall that holds the chain in place”.
Another wrote that, “I once came home and found my kitten hanging like that by the neck. I arrived in time to save him, but it was rather scary”.
In response to the above comment another commenter wrote, “My brother happened to walk out of his bed to pee at 1.30 a.m. and heard something scratching the blinds. Took a close look and found his roommate’s kitten with the cords around her neck. The luck of it all is absurd”.
“I came home yesterday from work to find one of my dog’s collars hanging from the blinds in the front window. He managed to get the tags all tangled in the twine that holds the blinds up. If he didn’t wiggle free he would have hung himself and I would have found him dead. I was so upset yesterday that I cried and vomited.”
I bumped into these comments very easily on a social media website. They indicate to me that it is not uncommon especially for young cats and dogs to play around with the cords of blinds, perhaps because they dangle down and kittens like to play with dangling things and then eventually end up becoming entangled in them and unable to release themselves at which point they are in danger of being strangled.