The whole "humans use 10% of their brains" trope is rightly called a myth but it's interesting to consider how it came about.
I think the main (or only) source of this conclusion is the observation that human brains (like all mammalian brains and probably most vertebrate brains) can sustain what seem to be nearly catastrophic losses while retaining almost normal function. There are many examples but the ones I know best come from neurodegenerative diseases. In most if not all cases (but I'm thinking of Alzheimer's, Parkinson's, and ALS), loss of neurons proceeds for years and results in huge reductions in cell number before clinical symptoms are recorded. In Parkinson's, for example, it was claimed for many years that it takes the loss of up to 90% of the dopamine neurons to cause deficits in function. I suspect that this kind of claim (now known to be incorrect) helped bolster the 10% myth.
Al (@aleo) provides a single strange case study of hydrocephalus. One can find any number of scenarios, involving disease or trauma or genetics, in which some part of the brain is decimated but the person (or other animal) seems to get by just fine.
Perhaps another major motivation for embracing this myth is the fact that we humans know about other humans who can do so much more than we can, perhaps despite significant limitations, and so we conclude that our own capacity is somehow untapped.
There are at least three problems with the reasoning here. The first one is obvious: it does not follow that because an animal can function with 10% of its hardware, it is not using the other 90%. People can see with one eye, but no one claims that the rest of us are only using 50% of our eyes. The second is more technical: reports of the ability of the human brain to withstand 90% neuronal loss have been discredited. The best example is the Parkinson's disease story: we now know that 30% loss of dopamine neurons is enough to trigger symptoms. Scientists are actively trying to understand the relationships between neuron number, synapse number, network wiring, and function. But it is clear that the 90% number, at least for PD, is way off. The third is the fact that we have watched human brains in action, and no one has seen anything resembling the 10% rule.
I do think we should all be inspired by those who overcome losses to do great things. And I like to be inspired by the fact that my brain is capable of greater things than I tend to ask of it. But the 10% myth is lame pseudoscience, and I would encourage everyone to use all 100% of their neurons to put it to rest.