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Kawoomba comments on Neil deGrasse Tyson on Cryonics - Less Wrong Discussion
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The brain has redundancy at the level of neurons: it is quite resilient against diffuse neuron loss, and in case of localized damage, unless the affected area is large or includes key regions such as the brainstem, impairment is often limited to one or a few functions, and in some cases it even reorganizes to transfer the lost functions to other areas, partially recovering them.
However, there is no expectation that the brain has redundancy against the loss of an information storage medium that is used in all neurons.
If you destroy half of your collection of DVDs, the information in the other half is still intact. If you destroy every odd-numbered track on all of your DVDs, instead, most of the remaining data will be too fragmentary to be of any use, even if the number of bits you destroyed is the same in both cases.
Depends on what axis of resilience (as you alluded to).
For memory, confer grandmother cells.
That wiki article looks dated. See these two, more recent abstracts: [1], [2].
Anyways, the point isn't whether there are actual grandmother cells, or "merely" a very small number of cells serving the same purpose. It is that there are crucial brain functions with little to no redundancy.