The better you know the material, the better you will know when to contest a ruling or contest validity. You should also know the rules fairly well. When another team is answering the question, you should never tune out. Pretend that you are the quizmaster and follow their answer against what you have memorized. Listen to the question and try to decide if it violates one of the rules for interrupted questions. In this regard, you do not have the benefit of seeing what the actual question is, however the rule of thumb is that when the quizmaster rules a quizzer correct, their question is effectively the official question. This means that if there is some difference between the completion and the question on their page, the quizmaster and/or judges have made the decision that it is the “same basic question”. Also, listen to their answer, making note of any differences between what the quizzer says and what the Scripture says. At the end make a ruling in your mind. If the quizmaster rules differently, you should consider a contest. Watch to see if the quizmaster and/or judges circle or write anything on their page. If they do, you know there is some difference. This can be your hint that there might be something to contest. If a quizmaster goes to the judges this means that there is something iffy and you should also consider contesting.
All of this must be mitigated by the fact that some decisions are simply judgment calls. Some quizmasters go to the judges any time they think there might be something that makes a quote wrong. Judges make a split-second decision as to whether it is correct or incorrect. It is usually rare for a team to contest a ruling on a quote and get it overturned. The only instance in which I have seen this happen is when the judges have actually made a genuine mistake, either by forgetting it was a quote or not knowing the rules well enough. For many other issues it is a complete judgment call. By this, I mean that there is no black & white ruling available and that if you were to present the contest in 100 quiz rooms, 50 would accept and 50 would deny. You should also keep in mind that when a quizmaster and/or judges make a ruling, they are going from the limited information that the question writer has provided on their page. Most officials have not memorized the material therefore do not know if a quizzer came from a “different” verse when the answer is a complete answer.