If you do it right, yes. Wikipedia is licensed under the same license as Stack Exchange content is: Creative Commons Attribution ShareAlike.
From the Wikipedia attribution requirements page, with a bit of reformatting:
Attribution. To re-distribute text on Wikipedia in any form, provide credit to the authors either by including:
a hyperlink (where possible) or URL to the page or pages you are re-using
a hyperlink (where possible) or URL to an alternative, stable online copy which is freely accessible, which conforms with the license, and which provides credit to the authors in a manner equivalent to the credit given on this website, or
a list of all authors. (Any list of authors may be filtered to exclude very small or irrelevant contributions.)
This applies to text developed by the Wikipedia community. Text from external sources may attach additional attribution requirements to the work, which should be indicated on an article's face or on its talk page. For example, a page may have a banner or other notation indicating that some or all of its content was originally published somewhere else. Where such notations are visible in the page itself, they should generally be preserved by re-users.
Copyleft/Share Alike. If you make modifications or additions to the page you re-use, you must license them under the Creative Commons Attribution-Share-Alike License 3.0 or later.
Indicate changes. If you make modifications or additions, you must indicate in a reasonable fashion that the original work has been modified. If you are re-using the page in a wiki, for example, indicating this in the page history is sufficient.
Licensing notice. Each copy or modified version that you distribute must include a licensing notice stating that the work is released under CC-BY-SA and either a) a hyperlink or URL to the text of the license or b) a copy of the license. For this purpose, a suitable URL is: http://creativecommons.org/licenses/by-sa/3.0/
Thus:
For tag wikis, you can merely copy and paste, but then you have to give attribution to Wikipedia in the form of a link to the source. Since Wikipedia has no rel attribute requirements (StackExchange does), there should be no problem.
Additionally, if you do copy, you must do so verbatim, possibly in the form of a citation. Wikipedia requires CC-BY-SA-3.0 or later for the remix right to be applied, but Stack Exchange uses CC-BY-SA-2.5!
For tag wiki excerpts, you can't. You can't add any formatting or links to tag excerpts and the list of authors would be longer than the excerpt itself.
In practice, it's much simpler if you don't do that.
You should still be free to collect information and rephrase it originally — only the form of facts can be copyrighted, the facts themselves cannot. However, I'm not a lawyer.
