The discussions here are very long, so I thought I'd relay this experience when I tried to top up my cellphone online on the TIM website (while in Rome). I was trying to do this because I had only a few Euros left and we were entering a holiday.

Bottom line --- UNLESS you are a resident of Italy and have a credit card with an Italian address, you cannot top up your cellphone on the TIM website.

Thanks for checking and clarifying. This is useful.