Situations when we cannot use confidence intervals in hypothesis testing? It seams like confidence intervals resemble critical regions in hypothesis testing .. are there cases when one cannot use these intervals in testing?
In critical regions we use the mean and see where it lands while in confidence intervals we use the mean to construct the boundaries.

Yes, there are situations where tests cannot be executed through confidence sets. Generally speaking, this happens in inferential problems where interest is not on a (sub)set of parameters but on the model itself. Goodness-of-fit tests are one such example.
For instance, when in a linear regression model we want to test if the residuals are normally distributed. The typical procedure for the test is:

*

*compute the observed statistics and the associate $p$-value, then compare the $p$-value with the type I error $\alpha$, or

*compute the observed statistic and compare it with the threshold value obtained from the (typically) asymptotic distribution of the chosen test.

As far as I know, there is no confidence set procedure that can tell us if the residuals are normal or not.
UPDATE
@kjetil b halvorsen points to this confidence set procedure obtained by inverting the Kolmogorov-Smirnov test statistic.
