There are restrictions on bag sizes, and you can find those here:
http://www.coopculture.it/en/heritage.cfm?id=1
"STRICTLY FORBIDDEN to introduce inside the excavations of Pompeii, Herculaneum, Oplontis, Stabia and the Museum of Boscoreale, handbags, backpacks, luggage, cases, larger than 30x30x15 cm." (approx. 12"x12"x6").
A word of warning about fannypacks? They aren't pickpocket-proof, even worn facing front. Keep that in mind when doing any sightseeing on your own that may involve public transport or other potentially crowded areas. Stow valuables (credit cards, passports, large amount of cash) beneath your clothing in a moneybelt or other type of security device.