Is there a typical example of Nisnevich covers?

There is a popular (and I think helpful) example of etale covers, namely covers of Riemann surfaces with ramification points removed. Is there a similarly accessible example to motivate Nisnevich covers?

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Personally, I have found examples of Nisnevich covers un-enlightening. The most important property is: $p: U \to X$ is a Nisnevich cover if and only if it is an étale cover, and there exists a sequence $X \supset Z_1 \supset Z_2 \supset ... \supset Z_n = \varnothing$ of closed subschemes such that $p$ has a section over each $Z_i - Z_{i + 1}$. This allows induction arguments. And of course another motivation for Nisnevich is that every Nisnevich cover of a field has a section, and so the cohomological dimension of a "point" $Spec(k)$ is zero - a "failing" of the étale topology. –  name Jul 27 '12 at 1:50
See my question mathoverflow.net/questions/103257/… for something I would have just emailed you. –  David Roberts Jul 27 '12 at 2:42