When talking about the Chebyshev's Inequality $$\Bbb P\left(|\hat{\theta}_n-\theta|\geqslant k\right)\leqslant\dfrac{Var(\hat{\theta}_n)}{k^2},$$ my instructor commented that it gives very conservative estimates. This means that the real bound of $\Bbb P\left(|\hat{\theta}_n-\theta|\geqslant k\right)$ is likely to be significantly smaller than $\dfrac{Var(\hat{\theta}_n)}{k^2}$.
Why is that the case? And why does the Central Limit Theorem give a more accurate estimate?