There are a few factors that can cause an IQ to be measured lower than the person's actual ability. In general, health at the time of testing, fatigue level, anxiety, attention span, the side-effects of medication, and other similar factors can lower scores. If someone needs glasses, doesn't wear them to the test and doesn't mention this to the examiner, their score can go down! In my clinic experience, I have delayed testing for children who had fevers or were on cold medicines when they arrived. I also delayed testing a child whose grandfather had died that morning. If you are going to the trouble of having psychological testing done, it really should be under optimal conditions for the person being tested.
There are other complicating factors. There has been a lot of debate about the validity of standardized tests for intelligence for minority groups and people who have had disadvantaged educational opportunities and a disadvantaged homelife. While I believe that there has been some improvement in the newer standardized tests for intelligence, it could still be possible for someone in this situation to have a higher ability than their full scale score indicates.
It is also possible for persons who once scored very high on an intelligence test to have a lower score when re-examined years later. Age, a progressive illness such as Alzheimer's Disease, or quite simply not using one's abilities can cause intelligence potential to decrease. I have come across students in my clinic experience who scored very high at a young age, but due to long-term underachievement, or not being challenged academically, show a drop in their scores to a more average range.
People are also not always uniformly gifted in their abilities. A person may be a gifted writer but have average mathematical skills (and vice versa). Psychologist Dr. Howard Gardner's theory of Multiple Intelligences states that a person can be gifted in one or more of several areas: linguistic, musical, mathematical, spatial, bodily (such as a great athlete or dancer), and interpersonal intelligences. Following his theory, a person could have gifted musical ability and average academic performance.
So to sum up, I think the answer to your question is yes, a gifted person might have an average IQ depending on how the giftedness is being measured.