It is NOT possible to "break free of" gravitational attraction, as it is infinite in range. However, it is possible to get sufficiently far away from an object that its gravitational influence on other objects becomes negligible, but it certainly cannot be zero. Note that I am talking about only two objects here. In multi particle system, an object can experience a zero NET gravitational attraction.
Textbooks that use the language of "escaping a planet's gravity" do a huge disservice to students and beginners for two reasons. The first is that "gravity" does not exist. Over 19 years of teaching undergraduate physics, I have come to realize that the word "gravity" connotes, in the minds of students, a material substance. Objects that don't possess this substance don't exert gravitational attraction. The second is that, as I said above, gravitational forces have infinite range.
The term "escape velocity" is also a misnomer because "escape speed" doesn't depend on direction. The best way to think of this is in terms of energy. The correct framing of the question is this:
What kinetic energy must I give a spacecraft so that by the time the spacecraft is "very far away" from a planet, it (the craft) has no leftover kinetic energy?
Of course, a speed can be inferred from the initial kinetic energy. For objects launched from Earth's surface, the required "no kinetic energy when very far away speed" is about 11 km/s.
Note also that Nic's answer refers to "the gravitational potential energy of a particle" which constitutes another physics error. A particle cannot possess any form of potential energy because potential energy is a property of an interacting system. Instead, we should define a system to consist of particle plus Earth, and THEN talk about the interaction, or potential, energy of that system. Then everything makes sense and is internally consistent and correct.