Yes, it is possible. It may be related with your habit of using eyes. If you are reading lying on the bed, you may look at your book with head tilted. Thus one eye will use overtime. That is the reason why one eye get dry first. However, you should treat your eyes carefully by using the eyedrops. If you don't want another one get infected, too. You'd better treat the dry eye first.
Yes, I think it's normal to have dry eye in only one eye. As we know that there are many reasons which can cause the dry eye. For example, you may get it , when you work too hard and have no rest. In this situation, you may have both of the eyes become dry eyes. But on the other hand, if your eyes are infected by some dirty things, it will just irritate one eye, at this moment, you should do a quick treatment, so that you can avoid the other one get dry eye. Or you can go and see a doctor as soon as possible.