Bounded Input Bounded State stability

Question

For a state space system $$\dot{\vec{x}} = A \vec{x} + B \vec{u}$$ I understand that if the real components of all eigenvalues of the matrix A are $< 0$, then the system is bounded input bounded state stable. I also understand that if the system does have a real component $\geqslant 0$, it doesn't necessarily mean that the system is unstable. My question is, can you conclude that an unstable system always has at least one real component $\geqslant 0$?