Some potentials have only finitely many bound states (the finite square and delta function are two good examples) Others have infinitely many bound states (for example the infinite square well and $1/r$ hydrogen atom potential). What is a condition that guarantees a potential will have infinitely many bound states?
A partial answer: The first chapter of Chadan and Sabatier’s ‘Inverse problems in quantum scattering theory’ explicitly defines so-called regular potentials
...to avoid the occurrence of some unnecessary complications (infinitely bound states etc). [emphasis added]
If we consider the scattering of a particle of mass m off a central potential, regular potentials are those that satisfy
$$\int_b^\infty |V(r)|r\,\mathrm dr < \infty$$ for $b \geq 0.$
So this is sufficient to avoid an infinite number of bound states for central potentials.