Many cultures can be somewhat protective of their traditional foods, and since Italian peoples' taste for cheese tends to be more moderate than most Americans', it can be a social faux pas to ask for additional cheese to be added to a dish that already has it. This applies especially to asking for extra parmesan on pizza.
Some dishes, especially those with seafood, are never served with cheese, and asking for cheese to be added can be seen as an insult. This is not to say that it's impossible to get extra cheese in Italy, but asking for more can seem boorish. Diners who don't want to seem rude may need to be happy with the cheese they're given.