Those large lever type bead breakers have the same problems the ones on actual tire machines have when trying to break down damaged, old tires. They don't really get between the bead and rim, so more often than not, when dealing with old, damaged tires, they just end up pushing the sidewall in and don't move the bead. The other problem is they apply pressure to one side of the tire/wheel, but don't secure the other side, which means the wheel just moves, not the bead.
The trick Gabriel mentioned, about leaving some pressure in the tire works really well, but, on damaged tires, that's not possible. I've changed a good many blown out tires, that's where something else can make it easier.
What needs pointed out about the product shortline10 mentioned, is that is breaks the bead by clamping to and applying force against the wheel itself. It holds itself in place, so it applies force directly to the bead and stays put while doing so. It's nothing like the large lever type bead breakers that apply force against the entire wheel/tires assembly, which tends to leverage the opposite side of the assembly up, consequently rolling the sidewall instead of staying put on the bead.
The story of three wheels and a man...