Not true.
I knew two people who actively used suicide threats to manipulate and emotionally blackmail people who did eventually kill themselves.
One of them did it whilst looking after his 18 month old daughter - stepped out leaving the baby with his drunk (shitfaced!) father, saying he was going to the loo - snuck out the back door having sent a FB message 'xnamex out', following threats to his ex girlfriend, kids mum that he was going to do it if she didn't drop off money (for the things he should have been buying for the baby out of his own funds).
She had called police, they were slow to respond as it was busy and this was not the first, second or even third time he'd done this (not blaming them in the slightest for that).
His dad woke up, baby screaming, cold and thirsty and hungry and with a sagging nappy... went out to see if he was having a smoke, carrying the baby - found him swinging from a tree.
The police arrived a few minutes later, to find him standing in the garden howling down the phone to the ex blaming her - still, all in front of the small child.
Can you imagine the guilt she'd live with if she HADN'T called the police?
The other case jumped off a motorway bridge, this was before the days of everyone having phones, she expected someone to find her note fairly quickly, as they had before, and go to find her at the usual spot. They didn't, she jumped onto the M66 and her parents arrived home to find the police waiting for them.
It is not OK to tell people 'oh the ones who use suicide as a threat to manipulate you will never actually do it'. It isn't true.