Your caveat about not being a national of either country is a bit puzzling, because your question is about acquiring the countries' nationality, which implies as a matter of course that you do not presently have either nationality. After acquiring each country's nationality, of course, you will be a national of that country, so by the time you are a dual citizen of Italy and the US, it will no longer be true that you are not a national of either country.
In other words, it's analogous to asking "Will I be able to get a driver's license after I learn how to drive? The problem is that I don't have a driver's license."
US law does not require you to renounce Italian citizenship if you naturalize as a US citizen. I don't know Italian law on the matter very well, but the relevant section in Wikipedia says, without citations, that naturalizing elsewhere does not cause loss of Italian citizenship. Assuming that is true, and that neither country makes any relevant changes in its nationality law, then the answer to your question is yes: you can be a dual citizen of both Italy and the United States.
(In fact, you may at that point hold three or more citizenships if your current country or countries of citizenship does not or do not have laws causing you to lose citizenship when you naturalize in Italy or the US.)