Suppose that $T_1$ is a $\alpha*100\%$ lower confidence limit for $\theta$ and $T_2$ is a $\alpha*100\%$ upper confidence limit for $\theta$.Further assume that $P(T_1<T_2)=1$. Find a $(2\alpha-1)*100\%$ confidence interval for $\theta$
I thought that $(T_1,T_2)$ was a $ {\alpha}^2*100\%$ confidence interval, but it is only if $T_1,T_2$ are independents (correct?)
Your statement about $\alpha^2$ looks peculiar given that you know $P(T_1<T_2)=1$.
The events $T_1 \gt \theta$ and $T_2 \lt \theta$ are not independent, since if $T_1 \gt \theta$ then $\Pr(T_2 \lt \theta)=0$. Indeed $T_1 \gt \theta$ and $T_2 \lt \theta$ are almost surely mutually exclusive
Hint: You should reconsider $\Pr(T_1 \le \theta \le T_2)$ in the light of this