That's the title of an article in the April 2006 Reason (not online yet, apparently), and its premise is that, despite claims to the contrary, "Americans are more likely than ever to stay put." I looked up the census data that the article points to, and it seems to support the article's assertion: In 1947-48, the first year for which data was systematically gathered, 20.2% of all Americans had moved; in 2002-03, that percentage was 14.2%, the lowest it had been since 1947-48 (except for 2000-01, when it was also 14.2%).
My one concern is that the Census data shows an oddly large and sustained dip from 1999-2000 to 2000-01, which makes me wonder whether there that dip reflects a change in data gathering rather than in the underlying reality. Nonetheless, even if we set aside that 2% decline, the mobility rate would still be lower today than in the past.
If anyone has more data on this, or more information on the 1999-2000/2000-2001 dip, please post it in the comments.