A person can certainly be efficient and ineffective. This can occur when the activity focused on for efficiency is not a proper priority or when efficiency is not the quality needed in the situation. A few examples come to mind.
First, let us suppose that a person is charged with running the human resources department. This person might be quite efficient in developing policies and procedures and distributing them to the workforce. But the policies and procedures might be completely ineffective. The first priority is the effectiveness of policy and procedure, not efficiency in creating and distributing them.
Second, I have seen people who are quite efficient in dealing with paperwork, following that great rule: Do not handle a piece of paper more than once. What happens is that this person deals with his or her in-box quite efficiently, dealing with everything that comes across the desk promptly. However, there are some times when that rule is made to be broken. Good decision-making is not always an efficient process, I don't think. It can be a messy process, thinking and gathering information and then thinking some more. Is the priority to keep the in-box clean or to make good decisions? This is where efficiency is ineffective.
There are many instances in which people pride themselves on their efficiency and are completely ineffective. Which is more important is situational, and all of us need both kinds of skills.