I'm not sure that's actually true. Yes, the govt. reported "official" unemployment number is 3.8% or whatever. BUT... that comes with a couple of big, big caveats:
1. Those numbers by definition don't include job-seekers who have left the market and quit being job-seekers. IOW, people who became so despondent that they gave up. But, there's nothing specific that stops these people (or some portion of them) from re-entering the job market to fill demand.
2. These numbers don't reflect underemployment where someone has "a job" but the job doesn't require the skills the individual actually possesses and pays significantly less than they would expect to earn in a "normal" position.
3. Wage growth is still very low, which suggests that the economy is not being hamstrung (yet) by lack of available workers. At some point, if demand exceeds supply by enough, you're inevitably going to see wage growth. A significant jump in wages would, IMO, be the best sign that there's an actual gap between labor supply and labor demand.
Note that I'm not saying that Bernanke's theory is wrong, but this one little point jumped out to be as something that's at least somewhat questionable.