I have many pairs of measurements. A paired-samples t-test on these produces a 2-tailed p > .1 (so non-significant by typical alpha values).
When I run two separate one-sample t-tests comparing against some common null (say 0), one such test produces a p ~= .01 (therefore significant), while the other produces p ~= .6 (therefore non-significant).
The 95% confidence intervals of the two measurements overlap substantially. The non-significant measurement's 95CI additionally overlaps 0, whereas the significant measurement's 95CI does not overlap 0.
So whereas the pairwise differences between the measurements are too small to reach significance, one is significantly different from 0 overall while the other is not.
There seems to be a logical contradiction between saying that the two measurements do not differ, when they differ in their proximity to a common value.