Many mechanical watchmakers advertised their movements as “antimagnetic”. This means that the balance wheel spring is not magnetic-a spring made from a magnetic material could be damaged by a strong magnetic field (presumably warped or torn loose).
Now, unless you are a particle physicist (working near a huge electromagnet), or a power station engineer (working near large transformers)-how likely would you be to encounter a magnetic field strong enough to damage a watch?
Was this mostly an advertising gimmick?
Or did lots of people in the 1950’s lose watches to unknown magnetic fields?
You are exposed to a magnetic field constantly - the natural one generated by the Earth.
Presumably, even a small field could cause errors in a timepiece.
Just a WAG, but it could be that it’s a gimmick aimed at those people who are convinced that they emanate some vague magnetic field that destroys every watch they wear.
I had a couple of inexpensive Timex mechanical watches in the 1970s and 1980s. These were wind-up watches (not automatic).
Both watches would stop running if I put an ordinary magnet close to them, and restart by themselves if I moved the magnet away.
I would guess that, depending on your occupation, if you worked near magnets a few times a day, you might need an “antimagnetic” movement or your watch would run late.
Don’t forget MRI machines, as one let his wallet get too close to an MRI machine and had to have all of his magnetic stripe credit cards be reissued…
J.
Oh, jeez. You mean it’s not just my mum?
True enough, though, I suspect that the typical 1950s wristwatch wearer didn’t often encounter MRI machines. 
I have an Omega Speedmaster Automatic–a fairly nice watch–and managed to affect it with a small neodymium magnet. It didn’t stop, but it did start losing several minutes a month. No permanent damage, though–I sent it in and they adjusted it back to normal.
So it doesn’t take an MRI machine, just a small toy magnet at close range. I’ve since been more careful when playing with magnets.
See ISO 764 - a bargain at only CHF 50,00.
I suspect the OP already knows of the Rolex Milgauss. Wiki says it was “specifically for those who worked in power plants, medical facilities and research labs (like those in Geneva) where electromagnetic fields can cause havoc with the timing of a watch” - but gives no indication of how significant a problem that really was.