Assume your hypothesis test concerns whether a certain effect or parameter is 0. With interval estimation, you can distinguish several options:

1. The effect is precisely-measured and the interval includes 0, so we can ignore it.
2. The effect is precisely-measured but the interval doesn’t include 0, but it’s close enough to be negligible for practical purposes, so we can ignore it.
3. The effect is precisely-measured to be far from 0, so we can keep it as is.
4. The effect is poorly-measured but we’re confident it’s not 0, so we can keep it but should still get more data to raise precision.
5. The effect is poorly-measured and might be 0, so we definitely need more data before deciding what to do.

…as illustrated below:

Imagine you’re a scientist, making inferences about how the world works; or an engineer, building a tool that relies on knowing the size of these effects. You would like to distinguish between (1&2) vs. (3) vs. (4&5). Journal readers would like you to publish results for cases (1&2) or (3), and should want you to collect more data before publishing in cases (4&5).

Finally, this is an issue whether your hypothesis tests are Frequentist or Bayesian. As John Kruschke’s excellent book on Doing Bayesian Data Analysis points out, if you do a Bayesian model comparison of one model with a spiked prior at $\theta=0$ and another with some diffuse prior for $\theta$, “all that this model comparison tells us is which of two unbelievable models is less unbelievable. [And] that is not all we want to know, because usually we also want to know what [parameter] values are credible” (p.427).