As Dario explained, the answer is no. Trains don’t sell out seats. Seats are always available even at the last minute.
Occasionally, but very rarely, the seats in the standard class (the cheapest class) sell out in some trains. In that extremely rare event, you either buy a ticket in a more expensive class on the same train, or wait for the next train, which, almost invariably, will have availability in all classes.
For example, the next train from Florence to Naples, departing in a few minutes from now, still has availability in standard class.
The advantage of booking ahead is getting a discount. However in that case you are locked on that specific train you book, as discounted tickets aren’t refundable or changeable.