Yes! Reserved trains can sell out! Once in mid-December I sauntered into Napoli Centrale at Noon on a Sunday, thinking I could buy one reserved seat to Rome easy for that day. Wrong! Not one reserved seat available. I had to wait for a "Special" train coming at 5 pm, which turned out to be a Regionale, standing room Only! With a transfer at Cassino - this station had No elevators, No escalators, No ramps, just those grand ballroom stairways, and No platforms. We all waited in a long line out in a weed and gravel field for another Regionale!
It took seven (7) hours to reach Rome. My lower back felt every lurch. Then at Termini, the train stopped at a staggered spot, meaning a stopped train was in front of us, giving us a mile or more walk to the station!
So, soon as I arrive at a station, I buy my reserved tickets out, wherever I'm going in the area and back home. Or if I'm Not sure of my departure day and time, I hightail it back to the station as fast as I can get there and buy my ticket back home. I think the latest I've bought tickets out was three days before departure, which is still cutting it too close in my experience. I try to buy all the tickets I need as soon as I arrive at the station.
You're competing with tourists, workers, students, and soldiers. In my case, you'd be surprised at how many Napoli work in Rome! A Lot! You've got students and soldiers going to Rome the night before so they can be available early Monday morning, as an example. I had stood for seven hours, I hobbled off the train and to the station, I was the last to reach the station, and Termini looked mighty good to me!