I'm flying on separate tickets from Bari to Rome (ITA) and then Rome to Philly (AA) because I used FF miles to book the AA flight. I will need to check my bags. So I will have to collect the bags in Rome and then recheck them in Rome for the flight into Philly. I have exactly 2.5 hours connection time. Will I make it?
If everything works perfectly you could make it.
However it is not something I would do. If anything goes wrong you could easily miss your flight. That includes a delay arriving from Bari, baggage slow to appear, long lines to check your bag again, slow security, slow exit immigration, or many other issues.
I would take a flight to Rome the night before and sleep at the airport if you have a morning departure.
I think it is too tight since you are checking bags and you are hoping that your first flight will be on time. I'm doing something similar next year but I will arrive the day before in Rome.