Air France bumped us even though we had already checked in. Claimed
the plane was too heavy due to wind. Then ended up bussing us to
Bologne for another flight. Apparently this happens quite frequently
there.
Yes. That happens whenever wind exceeds 10 knots.
FLR has a very short runway (5,100 feet) and due to the presence of a mountain on the northern end (Mt. Morello) airplanes can only land from the south and take off to the south. If there are tail winds exceeding 10 knots, the aircrafts can operate only with weight limitations (or not at all).
It almost happened to me last April (but the captain gave the ok to load the plane), and once in the 1990's (flight cancelled, this time due to fog, and rebooked from Bologna the next day).
That is a risk one runs using FLR. Obviously when it happens the airline will rebook you on the next available flight, but that might mean a full day delay if you intend to catch a connection at some other European airport. These delays are less common in summer, when winds in Florence are largely calm.
Unfortunately the new runway project (ready for years) is still stuck in the courts due to NIMBY's opposition to the project. They say work on a new modified project should start in 2024, but so they said in 2013, when they were supposed to be finished by 2017 before they took the case to court.