Breaking into top schools coming from a lesser known uni.

Hi guys. I am an Italian student studying finance at the University of Florence (Italy) and my dream is to break into IB. My plan is to do so after attaining a master in finance in a top tier European school. Unfortunately my uni is not as reputable as others in my country and I have always felt like I am at a disproportionate disadvantage because of that, consequently I have been grinding a lot to fill this chasm (4.0 GPA, starting my own club, learning new languages, semester abroad at ESSEC and what not). Quite sadly though, every time I hop on LinkedIn to scan some profiles of people who got the master I would like to one day have, from the schools I would like to attend (LSE, LBS to name a few), I notice that all of those with an italian degree come from Bocconi university. So my question is, is it possible to break into one of this schools coming from a less renowned uni? Thanks In advance.