I tested this a bit further and found out a bit more. MapDocument.saveACopy seems to embed some attribute with the output filename in both the MapDocument object and the resulting mxd file. MapDocument.saveACopy fails when writing to this same filename. Alternating filenames works by alternating the embedded attribute in the MapDocument object.
I apparently created my test.mxd by saving a blank mxd, at some point calling mxd.saveACopy("C:/tmp/test2.mxd"), and later renaming it to test.mxd. This explains why I only see problems with C:/tmp/test2.mxd and no other path, and only with this mxd.
BTW, I'm using Arc 10.1 SP1 and python 2.7.2. ArcMap's Save, Save As, and Save A Copy do not seem to do this.