Let's try a specific example. Look at the finite square well case with a depth of V_0. When you solve for the eigen energies for this case, you get a series of values that depends on 1/L, where L is the width of the well. It means that if you make the well narrower (L getting smaller), the magnitude of the ground state energy is now larger, i.e. the energy for n=1 state will be greater. At some point, the potential well is so narrow that E(n=1) = V_0, and the well can no longer have a state entirely within itself. So you will no longer have a bound state.
So the condition for a bound state depends not only on the depth of the potential well, but also on its physical size. To know rigorously if a potential well can have a bound state, you need to find out what is the energy of the lowest energy state it can have. If it is less than V_0, then you have a bound state. If not, you have no bound state.