Ken Jennings points out that today and tomorrow are both federal holidays (tomorrow only in the Washington, DC area), meaning that employees of the US federal government who work in the DC area will get the day off. Have there ever been two consecutive federal holidays before?
For those who don't know the schedule of holidays: the third Monday in January (that's today) is Martin Luther King, Jr. Day. January 20, in years with number one more than a multiple of four (that's tomorrow), is Inauguration Day. The reason that federal employees in the DC area get it off, if I understand correctly, is to keep the traffic down. (Not that it'll help tomorrow; from what I understand Washington will still be a mess.) As Ken Jennings points out, these days are consecutive if January 20 falls on a Sunday or a Tuesday.
Now, a presidential term is a whole number of weeks (208, to be exact) and five days long. (How do I know this? A common year is one day longer than a whole number of weeks; a leap year is two days longer than a whole number of weeks; thus a presidential term, consisting of three common years and a leap year, has five "extra" days.) So we can work backwards. The 2009 inauguration is on a Tuesday; the 2005 inauguration was on a Thursday. The 2001 inauguration was on a Saturday (which I could have told you anyway; I got in a car accident that day and remember the circumstances pretty well). 1997 was a Monday, 1993 was a Wednesday, 1989 was a Friday, and 1985 was a Sunday.
But Martin Luther King Day wasn't observed for the first time until 1986. So the answer is that Martin Luther King Day and the inauguration have never fallen on consecutive days. The pattern of when they do is kind of complicated, because leap years are periodic with period 400. But the 2013 inauguration falls on a Sunday; 2037 and 2041 are a Tuesday and Sunday, respectively; and most of the time these come in pairs; a Tuesday inauguration is followed by a Sunday inauguration. (The end of a century could break this pattern.)
However, the answer to Ken Jennings' actual question is yes, because of an obscure piece of trivia I just remembered: January 2, 2007 was a federal holiday, an official day of mourning for President Ford. (For some reason I remember checking my mail and being surprised there was none. This is strange, because there are lots of days where mail is delivered and I don't get any.) January 1, 2007, was of course New Year's Day. And December 31, 2006 was a Sunday, so there was actually no mail for three days.