There is definite truth in the notion that we do not use the full capacity of our cortex. It is generally accepted that there is a reserve present in the brain that can act as a backup for cerebral damage. Brain reserve can be defined in terms of the amount of damage that can be sustained before reaching a threshold for clinical expression (Stern, 2002).
A notable example is the removal of nearly half of the cortex (referred to as a hemispherectomy) in children with intractable epilepsy. Although behavioral changes may occur and motor skills may become compromised (Van Empelen et al., 2004), these children generally recover remarkably well and overall intellectual performance may in fact improve as compared to before surgery. (Note, however, that the quality of life and brain function of these patients was severely compromised by their medical condition) (Pulsifier et al., 2004). Another example where the brain is shown to have a reserve is the fact that cognitively normal elders sometimes are diagnosed with advanced Alzheimer’s disease pathology in their brains at death (Stern, 2002). In other words, while their cortex was severely damaged by Alzheimer's, they showed no clinical signs of cognitive deficits.
In all, the brain has a remarkable amount of reserve, which may be interpreted as an incomplete use of total brain capacity.
While this reserve is critical for brain plasticity in response to injury and aging, it is questionable whether it is available to enhance cognition as depicted in the movie.
- Pulsifier et al, Epilepsia (2004); 45(3):243-45
- Stern, J International Neuropsychol Soc (2002); 8: 448–60
- Van Empelen, Brain (2004); 127: 2071–79
The linked wire article in the accepted answer is popular science and I wouldn't base a conclusion on a '60-second-all-you-need-to-know' pseudo-scientific web link.