It's not uncommon to hear people use this familiar adage while surmising that full use would lead to telepathy, world peace, and the ability to fry soup cans with eyebeams. If only!
Well, it happens that a small percentage of the population have managed to use all of their brains all at once -- these people have experienced what is called a grand mal seizure (the kind we think of when we imagine epilepsy) and it's characterized by a loss of consciousness and violent bodily contractions.
Of course, it's not true that we only use 10% of our brains. We use all of it, but we selectively activate and inhibit certain parts as appropriate for getting things done.
For example, there's a part of thalamus (the VLo) that we want to keep turned off (inhibited) most of the time. This part of the thalamus transmits voluntary motor commands and we allow it to turn on exactly when we want to transmit them, the rest of the time we keep it turned off so that static firing doesn't send our muscles into wonky contractions. A loss of this inhibitory ability results in ballism, a serious condition involving violent and uncontrollable flinging movements.
To get any idea of just how active your brain is consider that the brain represents about 3% of your body's weight and uses almost 20% of your body's energy.