Yes, that's generally thought to be true — as determined through the science of modern food safety. You can read about specific guidelines on the Food-Safety tag wiki. Those practices were developed as broad recommendations, usually erring on the side of caution.
That being said, you might be able to get away with it sometimes, without dying of food poisoning, but it could be risky. Much depends on the food itself — bread left at room temperature may last for days, but you could have problems with meat or dairy products after just a couple of hours.
Bacteria and other harmful pathogens can exist everywhere. At room temperature, those organisms can multiply very rapidly, producing toxins that can make you sick. You can NOT rely on smell to detect them — food can be highly infected, but smell and taste just fine. That's why, for safety, it's important to refrigerate or freeze cooked food to slow down the growth of any pathogens.
It's best to do this as soon as possible after the food cools down to room temperature. You also want to avoid putting hot food directly into the refrigerator because it may heat up the 'fridge, affecting other food inside, and actually delay your food cooling down to a safe temperature.
Food poisoning isn't something that only breaks out when we grow old, though elderly people, young children, and those with weak immune systems can be more susceptible.
As leaving cooked food at room temperature was "normal" in your country, I wonder if there wasn't also a prevalence of food poisoning that was also just considered "normal'. How often did people have upset stomachs or diarrhea, and just accept it as part of everyday life?