# Failing to reject the null hypothesis

I have a very basic doubt but I am new to inference statistics so please pardon me. My question is when we do hypothesis testing for two samples, we estimate the confidence interval and then if the confidence interval includes 0, we fail to reject the null hypothesis (if I am not wrong?). Now as far as our critical regions are concerned, we fail to reject the null hypothesis if the t-score lies in the non-critical region. So then aren't these two ideas contrasting? What if our confidence interval consists 0 but then our t-score is in the non-critical region. Do we fail to reject the null or do we accept the null hypothesis?

Usually, test and confidence interval will match so that $p\leq \alpha$ corresponds to $1-\alpha$ confidence intervals (CIs) contains null value of the parameter of interest. This is for example usually the case for a single comparison for continuous data.