To make this even clearer:

Let's imagine that there is a huge amount of time dilation (we live on the surface of a neuron star or something). By climbing a bit, we experience 1.1 seconds instead of 1.0 seconds experienced by someone who left down.

We have a clock that can measure milliseconds as the smallest tick. But climbing up, back down, and comparing the amount of ticks won't let us conclude anything after a single millisecond. If anything, we must spend at least 11 milliseconds up to have a noticeable 11 to 10 millisecond difference.

Now, if the dilation was 1.01 seconds vs 1.00, we would need to spend at least 101 milliseconds up, to get a minimal comparison between 101 and 100 milliseconds.