I'll cite this example, not involving Apache, that I worked on a couple of years ago as a project for DISA in the Department of Defense. They had software that was written by U.S. federal government employees called CMIS. They wanted to release this software under an open source license so that it could benefit from wider adoption and encourage a larger development community both within the U.S. government and outside. They even wanted to encourage commercial ventures to take on this software and enhance it for all users.
We could have just taken it. The software was public domain, available if not automatically then at least under a formal Freedom of Information Act request. But instead, we negotiated a Cooperative Research and Development Agreement (CRADA), executed between DISA (DOD) and an outside non-profit open source foundation, the Open Source Software Institute (OSSI). That CRADA allowed OSSI to take over responsibility for the software; build a forge-like infrastructure for development purposes; and subsequently distribute the software to the world under the Open Software License (OSL 3.0). The U.S. government now uses the OSL version of the software rather than its previous public domain version.
When I mentioned earlier that we work with US government agencies "all the time", that may have been a slight exaggeration. I was referring to projects such as OODT, which was sponsored by NASA and conveyed to Apache. I don't know how much of that code was written by contractors and how much by government employees. That shouldn't be an important distinction, though, for those of us who want to transform such software into FOSS so that it can be distributed under recognized and defensible licenses. I also note that NASA has other projects it chooses to release under its own open source license, the NOSA. (http://ti.arc.nasa.gov/opensource/).
The National Institute of Standards and Technology worked with the OpenSSL project to create a government-certified version of the software. That indispensable program remained open source. (http://gcn.com/Articles/2006/07/18/Status-of-OpenSSL-FIPS-certification-shifts-again.aspx)
When the Department of Homeland Security develops advanced technology nowadays, it is choosing to do so under an open source regime. (http://www.oss-institute.org/index.php?option=com_content&view=article&id=422:host-2-demo&catid=141:dhs-host&Itemid=216)
These may not be so many examples as to justify my saying "all the time", but it is often enough nowadays to create a trend and to provide models that other U.S. federal government agencies can follow.