In a hotel with a finite number of rooms, it is clear that once it is full, no more guests can be accommodated.  Now, imagine a hotel with an infinite number of rooms.  One might assume that the same problem will arise when an infinite number of guests come along and all the rooms are occupied.  However, in an infinite hotel, the situations "every room is occupied" and "no more guests can be accommodated" do not turn out to be equivalent. There is a way to solve the problem: if you move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3, etc., you can fit the newcomer into room 1.  Unlike a finite hotel, in an infinite hotel, being "full" in the sense that every room contains a person is not the same as being "full" in the sense that there is no space for another person.

http://en.wikipedia.org/wiki/Hilbert's_paradox
_of_the_Grand_Hotel

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No room at Hendrick for Junior; could that change?

By Jenna Fryer, The Associated Press