Note the qualification "in an observational setting". 

While more context would be needed to be certain, I'm assuming the intent is "in the real world" rather than in simulations ... and in that case, perhaps it's a consequence of the fact that the assumptions under which the intervals are derived don't actually hold, so things that can impact bias - which are of small effect compared to variability in small samples - don't reduce in effect size as sample size increases, while the standard errors do.

Since our calculations don't incorporate the bias, as intervals shrink (as $1/\sqrt n$), the unchanging bias looms larger, leaving our intervals less and less likely to include the true value.


Here's an illustration - one which exaggerates the bias - to indicate what I think is meant about CI coverage probability shrinking as sample size increases:

![Diagram of CI coverage probability shrinking as sample size increases when bias is present][1]


  [1]: https://i.sstatic.net/bbjfq.png