I looked at my watch at 12:05. I wasn't sure, for a moment, whether it was 12:05 or 1:00; I had to carefully look to determine which of the two hands was the longer one.
A question for you: how many times in a given twelve-hour period could I have this problem? More rigorously, suppose I have an ordinary twelve-hour analog clock, with an hour hand and a minute hand but no second hand. Furthermore suppose I can measure the position of the hands absolutely precisely, and they're "sweep" hands (i. e. they move at a constant angular rate, without "ticks"). At how many times between (say) noon and midnight could I interchange the hands of the clock and still have the hands in a position that corresponds to some time -- but not the time that it actually is? Noon, for example, is not such a time; if I interchange the minute and hour hands at noon I get a valid position of the hands, but that's the position the corresponds to noon. (I won't give an example of a valid time because giving one would be a big hint.)
Bonus: what are these times?
Another bonus: Add a second hand; are there still times which give rise to ambiguous hand configurations? (I don't know the answer to this one.)
(No fair looking up a solution; this is actually a pretty well-known brainteaser. It's well-known enough that I probably knew it existed, somewhere in the back of my mind, before I reinvented it today.)
edit (1:14 pm): Boris points out that he wrote a very similar question as question 23 of this test (PDF).