Is there a sharp distinction between atoms and non-atoms? Suppose we have an electron and proton that are separate... not forming an atom... At some point when bringing them together, they form a hydrogen atom.
What is the dividing line between atom and non-atom and what exactly is the distinction? Is there a sharp difference in the nature of the interaction between an electron and proton when they form and atom and when they don't? What's the nature of this distinction?
 A: A hydrogen  atom is formally formed when an electron is sitting in an energy level of the corresponding potential.

A stable hydrogen atom is when the electron is sitting at the n=1 ground level. If it is in any other level the atom is unstable, and the electron will cascade down to the ground level. At the ionization energy, where n theoretically is infinity,a free electron nearing a free proton will cascade down with electromagnetic radiaton  to the ground level. Whether it will be caught, or a slight change in the energy will send it off and away from the proton is a matter of probabilities. 
In cosmology it is an important state when free electrons and protons start forming hydrogen atoms. It is a thermodynamic phenomenon, and is described here
A: The electron is defined to be "bound" to the proton if its total energy (kinetic + potential) is negative. In this case there is a maximum distance away from the proton that it can reach. It's very similar to planetary orbits - the electron needs to reach "escape velocity" (corresponding to zero energy) in order to, well, escape.
An atom is defined to be a nucleus and bound electrons. If you give the electron in a hydrogen atom the "ionization energy" for it to escape arbitrarily far from the proton, then the atom falls apart into a nucleus and a "loose" electron.
