I believe that what I am about to ask has a negative answer, but I can't seem to find a quick counterexample.
Consider a family of bounded operators $\{T_\alpha\}_{\alpha \in \mathcal{A}} \subset \mathcal{B}(X)$ where $X$ is a Banach space. The family has the property that for every $x \in X\setminus \{0\}$ there exists $M_x>0$ such that $\|T_\alpha x\| \geq M_x ,\ \forall \alpha \in \mathcal{A}$. Is it true then that there exists an $M>0$ such that $\|T_\alpha\| \geq M$?
I tried the approach used for proving the original Uniform Boundedness Principle, but it doesn't work.