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the question is - How many times in a day is it impossible to tell the time by a clock with identical hour and minute hands, provided we can always tell whether it is a.m. or p.m.? What happens if we cannot distinguish between a.m. and p.m.?
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2 Answers

When you know am or pm you can tell only at:
12:00 am
1:05:27
2:10:57
3:16:22
4:21:49
5:27:16
6:32:44
7:38:11
8:43:37
9:49:06
10:54:32
12:00 pm
1:05:27
2:10:57
3:16:22
4:21:49
5:27:16
6:32:44
7:38:11
8:43:37
9:49:06
10:54:32
12:00 am (repeat)
 
22 in 12:00 am ≤ t < 12:00 am.
 
When you don't know am or pm you never know the correct time.
 
I read "identical hour and minute hands" as being same length.  I also presume we cannot cheat and watch the hands over time to deduce which is hour and minute hand.
 
Therefore, the ONLY time when it is unambiguous is when the two hands are at identical angles wrt 12
 
This occurs 12 times each (am & pm).  For example ~1:06 and all subsequent overlaps
 
How many times in a day is it POSSIBLE!: 24  
 
!) (I changed it from impossible to avoid the continuum of answers.  If one considers dt to be arbitrarily small, (ie analog), then it becomes semantically difficult because the answer would be infinite.  If dt is granular (ie second or minute) the answer becomes a function of this granularity.  Instead, I simply state that 24 times are unambiguous if (am, pm are known)
 
2)If we cannot distinguish between am or pm, there are NO times where we are 100% confident

Comments

Rather than edit my answer, I'd like to acknowledge Steve S. as having the correct answer (22 and NOT 24 on part 1).  The 11th hour is missed and becomes the 12:00 spot which reduces each unambiguous time to 11 for each am & pm.  Good work Steve S ;-)

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