Sorry to resurrect an old thread, but it's the first result on Google when searching for BA FCO bag drop timings.

Just to add a datapoint: we arrived at the airport at 11:15 for our 16:45 flight (CE, GCH). Check-in agents were routinely asking people what time their flight was, and I was sure we'd be rumbled and asked to wait in the terminal. Instead, when I said 16:45 the agent said ok, and asked how many bags we had. Seems like if there is a policy about T-3 hours, it's selectively enforced - and we got lucky this time!