"Nerve signals are limited to the speed of light".
Erm, only in as much as they can never get near to the speed of light.
Nerve signals are actually limited by the speed of the electrochemical processes that are involved.
Nerves are not continuous fibres; they are broken up into "lengths", with tiny gaps in between. A tiny electrochemically produced current runs down the "length" of one part of the nerve, hits the end of the nerve, causes a chemical reaction (at the end of that "length" of nerve) that bridges the gap between the two "lengths" of the nerve. The chemcials cross the "gap" to the beginning end of the next "length" of nerve, which causes another electrochemical reaction that sends another small charge down the next "length" of nerve, and so on.
Admittedly, this all happens extremely fast, but the speed is nowhere near the speed of light. The same process occurs in the brain, on a bigger scale (due to the complexity and closeness of the nerves in the brain) but, again, the process doesn't achieve near-C speeds.
In my line of work (mental health nursing) we see direct evidence of these processes breaking down in the brain, to the extent that some patients can take incredibly long times to answer simple questions. They appear to "go blank" and there does not appear to be any conscious recognition of the question that has been asked. However, the patient has heard the question; it is just that they physically cannot respond in a timely manner.
Their physical movements are significantly slowed due to the deterioration in the electrochemical processes, resulting in reduced coordination of limbs, reduced speed of movement and action and a degree (in some cases) of muscle wasting.
These symptoms are commonly evident in severe cases of psychosis, untreated deep depression and many of the trauma-induced conditions (such as "shell shock", which is a hysterical conversion disorder that can result in an individual not being able to see, use a limb or walk, without evidence of physical injury).
The classic syndrome that displays all of these symptoms of the slowing of the body's electrochemical processes is the state of psychotic catatonia, where the individual is unable to move of their own free will. Their limbs can be placed in various positions and will remain in that position, until the position is changed by someone. The limbs or position can be held in that way for sometimes up to days at a time.