If a train has no more seats then it is sold out.
If it is a Regionale it can never sell out because seats are not reserved and there is no limit on tickets sold. If you can squeeze on you can go. Tickets do not vary in price and Trenitalia do not find those train prices more than 7 days out.
If it is a faster train - InterCity, EuroCity, Eurostar Italia, Freccia, all seats are reserved and no standing is permitted. Tickets for those trains are available much further out and are in typically three fare groups. The base fare is like the rack rate - for those who want ultimate flexibility or who haven't planned in time and are the highest fares. There are 2 levels of economy ticket also offered, which are very cheap and pretty cheap. Every train is allocated these - the seats are identical but the lower prices fill up first, leaving just the base fare until it too is gone - then the train is sold out.
If you are seeing trains after your wife would get out of bed which only offer the base fare then the discounted ones are all gone and will never be seen again for that particular run. What you see at the station will be the same but with the added fun of queues.
If you look at an upcoming train now and it has some super-economy seats left now you might look at it in 6 hours and see none. That's because in the interim the have sold those seats and now the cheapest tickets on this run are more expensive.