In linear control theory, a system is stable if and only if if satisfies the Routh–Hurwitz stability criterion, so we can use this to solve for the limits of stability. E.g. you can find the maximum gain that will allow for a stable system.
However, for nonlinear systems, if we can find a candidate function that satisfies the Lyapunov conditions, the system is stable, but the failure of a candidate function to satisfy the conditions doesn't mean it's unstable.
What is an example where one can find a Lyapunov candidate function that proves stability for a range of parameters, but where the system is still stable outside that range? I.e. where the stability bounds suggested by the Lyapunov function aren't the actual stability bounds?