Answer 1 - 'yes'
Suppose I ask, 'Can Tom walk ?' I am asking about the truth-value of the proposition, 'Tom can walk'. I expect the answer 'Yes' or 'No'. The answer, 'Yes', is right if Tom can walk and wrong if he can't - in which case the right answer is 'No'.
There are, of course, since we are using a natural language, indefinitely many possibilities for the actual meaning of 'Tom can walk' but this is not a wrecking problem. There is no reason in principle why we should not be able to agree on the contextual meaning of 'Tom can walk'.
Right, so I am saying 'yes', a question, this question, has only two answers. And (to work back to the terms of your heading) if the question actually only has two answers then it follows that it is possible that it has only two answers : actuality implies possibility.
You object :
"yes" implies that in a two-option system like the one above I can't give another answer, like: to leave the boxes blank, to fill both, to jump to my feet, etc.
But leaving the boxes blank, filling in both, jumping you your feet, are not answers - they are merely behavioural responses. If I ask whether you would like a cup of coffee, yes or no ?, and you throw the cup in my face, this would be an interesting response but not (without a lot of scene-setting, to which you do not appeal) an answer.
You reply to Eliran :
There's no written rule, moreover the rules could change or be differently interpreted. I understand what you mean, but what differentiates a reaction from an answer?
But you gave no indication at the start that while you were using 'question' in a standard way, no such restraint applied to your use of 'answer'. If the point of a question is to determine whether a certain proposition is true or false, your various behavioural responses - 'reactions' - determine no such thing and leave me none the wiser. Giving an answer is, of course, itself a behavioural response but the responses you list are truth-irrelevant.
Answer 2 - 'no'
If I say, 'no, it is not possible that a question has only two answers', but why should I?, what follows ?
it's possible that a question has only two answers.
I can't see how this follows. How can 'It is not possible that a question has only two answers' imply 'It is possible that a question has only two answers'.
A problem is clearly on your mind but I can't see what it is. I may well have misunderstood what's concerning you but in that case my answer may help spotlight the misconception.