From what I know and was also told by several doctors when a family member was dying of mesothelioma in his early 70's (despite never having worked more than casually a few times around asbestos in his much younger years), the illness is strongly suspected of being tied to a genetic predisposition that only affects a certain small percentage of people.

If one has this genetic disposition, it's apparently possible that even moderate or modest exposure early in life is enough to cause the (extremely deadly and almost impossible to treat) disease at some point later in life. This point can come years or decades after exposure.

And if one isn't predisposed, then even heavy exposure to asbestos probably won't cause mesothelioma. It is of course still possible to get other really shitty lung problems from the mineral regardless of genetic factors, but meso itself seems to be very selective. This apparently is why it's relatively rare despite widespread asbestos use for decades before recent years in much of the world.