Let $X_{1},X_{2},\ldots,X_{n}$ be $n$ random variables.
It's known that, $P(X_{i} > t) \le f(t)$, $\forall i=1,2,\ldots,n$ where $f(t)$ is a function of $t$ and is the same across all $i$.
Let $X^{*}=\max\{X_{1},X_{2},\ldots,X_{n}\}$.
Can I claim the following? $P(X^{*} > t) \le f(t)$
No, of course not. Consider Uniform[0,1] random variables, and $f(t) = 1-t$. It's easy to see $P(\max\{X_1,...,X_n\} > t) = 1 - t^n$, and so the statement fails.
If you want to reverse the inequality, $P(X_{i} > t) \ge f(t)$, then the reverse statement should be true.