If you are flying from FCO to NAP, you are certainly flying with ITA Airways (the "new" Alitalia), since it's the only airline covering that short segment.
So if you are flying from SEA to FCO with a code share with a sky team airline (e.g. Delta, AirFrance-KLM, ITA Airways) you can ask the check in person in SEA to tag your checked bag all the way to NAP. That way you don't need to worry about the checked luggage until you arrive in Naples. So no need to wait for the bag at the carousel in Rome. You will need however change flights in Rome. The time necessary to do so depends also on where your flight to Rome originates from. For example if you arrive from Paris (Delta and AirFrance both have non stop to CDG from SEA) or Amsterdam (Delta flies to AMS from SEA), then you will be arriving from a EU Schengen country, therefore you don't need to go through passport control (immigration) again. If instead you arrive to Rome from JFK or ATL with Delta or ITA Airways, you will need to clear passport control in Rome. Generally one hour is sufficient, but if you have to clear passport control and several international flights arrive simultaneously, there might be delays. In any case make sure that you ask the check in clerk at Seattle to tag your checked bag all the way to Naples (make sure the tag has NAP in it).