We consider bounded solutions of the semilinear heat equation $u_t=u_{xx}+f(u)$ on $\mathbb R$, where f is of a bistable type. We show that there always exist bounded solutions whose $\omega$-limit set with respect to the locally uniform convergence contains functions which are not steady states. For balanced bistable nonlinearities, there are examples of such solutions  with initial values $u(x,0)$  converging to  0 as $|x|\to\infty$. Our example with an unbalanced bistable nonlinearity shows that bounded solutions whose $\omega$-limit sets do not consist of steady states occur for a robust class of nonlinearities $f$.