Here we provide some examples of failures of the Binomial Assumptions, so that the result does not have a Binomial Distribution.
Example: Consider a students score on an MCQ test. For each question, the student gets a 1 or a 0 depending on his answer. The score is the sum of the scores, so it has a possibility of being Binomial.
1. If the questions are of varying levels of difficulty, then the probability of 1 will vary from question to question, being low for difficult questions, and high for easy questions. Then the sum of the Bernoullis will NOT be Binomial, because the p is not fixed.
2. Even if all the questions have the same level of difficulty, independence may not hold. Suppose that the first 10 questions are all about the quadratic equation. Any student who knows the quadratic equation can solve all of them, but he does not know then he cannot solve any of them. Then the student will either score 10 points or 0 points, but cannot score any points in between, so this will not be a Binomial distribution.
3. Another way that independence can fail is if the questions form a sequence. You have to find the answer to question 1 in order to be able to answer question 2 and so on. Then if someone gets a 0 on question 1, he MUST get a 0 on question 2, and so the scores are dependent. Again the Binomial model would not apply.
4. If some questions are weighted more heavily than others -- some questions have score 2, while others have 1 -- then again the Binomial Model will not apply
It is important to note that the Binomial Model may give satisfactory results, even if the assumptions fail -- satisfactory depends on the purpose for which the model is to be used.