Unless you have access to every student's score, you cannot know that the distribution is a normal distribution. It isn't clear that you do have access to the scores, but in a comment you say, "the 50 percent people were slightly less than the people who got 30-40 and 50-60 but not by much." You have described a bimodal distribution -- one with two humps at the ends and a dip in the middle. Your "slightly less" says the dip in the middle is a small one.
My own experience as a college teacher is that a bimodal distribution is a usual and expected result for a fair exam. At the "good" end, you find the people who either worked hard or found the material to be familiar. At the "bad" end, you find the people who did not work hard. That is exactly what you've described, although you equate "worked hard" with "had more time to work."
As far as interpreting the scores, you don't say whether the scores you mention are percentages, i.e. out of 100, or have some other base. If they're grades out of 100, then that is probably, but not certainly, a poor exam. (The other choice is that there's an entire class of poor students, and yes, that does happen.) If I, as a teacher, got a result like that, I'd use the grades as they were, but work much harder on the questions for the next exams.
Finally, you question whether the exam "was a good reflection of individual capabilities." That's not what en exam is supposed to measure. An exam is supposed to measure your knowledge of the course material. A very capable person who doesn't study can and should earn an unsatisfactory grade.