When Albert Einstein already was quite a famous Professor in Zurich,
he got on a tram with a friend of his. The conductor, who liked
brain-teasers, knew Einstein and asked him the following question:
"How many times per day is it possible to exchange (mentally) the hour
hand and the minute hand of a clock, and again get a possible
alignment of the hands (an alignment that shows a feasible position of
the hands)?"
Einstein continued his ride on the tram and the discussion with his
friend, but when they got off, he gave the conductor the correct
answer, also telling him he would have to wait for the proof until the
next day.

So the question is: "How many times per day is it possible to exchange
(mentally) the hour hand and the minute hand of a clock, and again get
a possible alignment of the hands (an alignment that shows a feasible
position of the hands)?"

If I understand correctly, for the time to be reasonable... then the
minute and hour hand would have to be pointing at exactly the same
place.  Otherwise, when you switch the hands then the hour hand will
be in the wrong place compared to where the minute hand is (ie the
hour hand may point directly at the 6 however the minute hand will not
be directly pointed at the 12).

This occrurs 22 times throughout the day.  1 time almost every hour
(approximately 0000, 0105, 0211, 0316....  but notice that it does not
happen during the 11th hour or the 23rd hour).

Since there are 24 hours in the day and 2 hours where this does not
occur, the answer is 22.  I could be thinking about the problem
incorrectly and I'd enjoy seeing my errors if so.

My initial guess is 286 times.  It occurs once in every five minute
interval all day long, except for two.  If we define a five minute
interval as including its beginning time but not its end time (so that
12:00 is included in the 12:00-12:05 interval, and not the 11:55-12:00
interval) then I think the only 5 minute interval during which
switching hands is never possible is 11:55-12:00 (I'm not sure of this
yet).

Here's one way to convince yourself that the hands can be switched at
other times besides when they're right on top of each other.  At some
time a little after 2:20 say, the hands will obviously be close to
reversed from where they are some time a little after 4:10.  So, set
two clocks to these times, and superpose them on top of each other. 
Grab the minute hand of the 4:10 clock and move it till it's perfectly
on top of the hour hand of the 2:20 clock.  The hour hand of the 4:10
clock will move a bit when you do this.  Now hold both the 4:10 minute
hand and the 2:20 hour hand and move them back and forth together
always keeping them directly on top of each other.  The 4:10 hour hand
will move in the same directions as you're moving your hand, but 12
times slower, and the 2:20 minute hand will move in the same direction
as you're moving your hand, but 12 times faster.  So as you shift the
4:10 minute hand and the 2:20 hour hand, the 2:20 minute hand will be
moving 144 times faster than the 4:10 hour hand, so the 2:20 minute
hand must catch up with the 4:10 hour hand.  At the point when this
occurs, the both hands on both clocks are perfectly aligned, and so
can be switched.

I stand corrected.  Once almost every 5 minutes (with the exception of
the 5 minutes between 55 and 60 in the 11th and 23rd hours).

24 hours a day times 12 sets of 5 minutes minus 2 exceptions
24 X 12 - 2 = 286

Einstein and Racecar can take the full credit for this one :)