Rome to Pisa to Interlaken

Afraid not. Pisa Italy is more than 300 miles away from Interlaken Switzerland.

Technically, it is possible. I checked the timetables on the German Rail site and entered a 2-hour stopover in Pisa. Leave Roma Termini at 06:57 and arrive at Pisa Centrale at 09:29. Leave Pisa Centrale at 11:40; connect at Milano Centrale, Brig, and Spiez; and arrive at Interlaken Ost at 20:28. You could have a 3-hour stopover in Pisa (with one additional connection) by leaving Roma Termini at 06:57 and arriving at Pisa Centrale at 09:29. Leave Pisa Centrale at 12:54; connect at Firenze S.M.N., Milano Centrale, Brig, and Spiez; and arrive at Interlaken Ost at 20:28.

Actually, it can be done. Leave Rome Termini between 6:57 and 9:15 am, allows 3 to 4 hour stop in Pisa, getting you to Interlaken between 8:30 and 10:40 PM. Some itineraries the last leg from Spiez to Interlaken is on a bus.
To view these, go to www.bahn.com, put in a near date and 6 am departure and add a stopover at Pisa Centrale for 3 hours. Pisa San Rossario is actually walking distance of the Leaning Tower (if that is the goal), but doesn't have luggage lockers. I don't know for sure if Centrale does. Maybe someone else knows for sure.

According to trenitalia.com, Pisa Centrale has left luggage facilities. The site doesn't say if there are coin-operated lockers or staffed services.

Where you leave your luggage at the Pisa train station is staffed. Allow enough time to retrieve it, factoring in having to wait for others to get their bags ahead of you, or if the staff person is on a break.