TL;DR: The property bounded, bounded from above, and bounded from below are different things, cf. Wikipedia.
In detail, consider a densely defined symmetric linear operator $A:D\subseteq H \to H$ in a complex Hilbert space $H$. Let $$\langle A \rangle_{\psi}~:=~ \frac{\langle \psi, A\psi\rangle}{||\psi||^2}$$ for $\psi\in D\backslash\{0\}$. It follows that $\langle A \rangle_{\psi}\in\mathbb{R}$ is real.
That $A$ is bounded from below means that $$\exists C\in \mathbb{R}~ \forall \psi\in D\backslash\{0\}: ~~ \langle A \rangle_{\psi}~\geq ~C. $$
That $A$ is bounded from above means that $$\exists C\in \mathbb{R}~ \forall \psi\in D\backslash\{0\}: ~~ \langle A \rangle_{\psi}~\leq ~C. $$
That $A$ is bounded means that $$\exists C\geq 0~ \forall \psi\in D\backslash\{0\}: ~~ ||A \psi||~\leq ~C||\psi||, $$$$\exists C\geq 0~ \forall \psi\in D\backslash\{0\}: ~~ \frac{||A \psi||}{||\psi||}~\leq ~C, $$ which is equivalent to $A^{\dagger}A$ $(=A^2)$ being bounded from above, which in turn is equivalent to $A$ being bounded from both above and below.