like a process failure? well, basically no, our brains are not binary systems, and even better, not 1 system.
To maintain your analogy, it is better to think of the brain as several computers each with different architectures and platforms, running different services and networked together, sharing information and, ultimately, coordinating the movements of a mechanical device it sits within.
The mammalian brain is, evolutionarily, several brains, each with certain, still nearly complete, autonomy. So, first, even if you were to completely disable one of these brains, you actually would not stop cognitive function, but there would be differences, some potentially not noticeable, some highly noticeable. In particular, removal of higher order (evolutionarily) brain can leave you with an apparently normal, but possibly impulsive and certainly less imaginative person. This is the frontal lobotomy.
Keeping with the IT systems, some seizures are more like broadcast storms, preventing the proper timing and signaling of services. Some seizures are like a system or service fault, where the system or service hangs, and the other, dependent services, have to adapt or error out. And some seizures are like massive electrical surges, forcing system resets of some or all systems.
In reality, when challenged to form a definition of a "system fault" it is very difficult for me to land on a satisfactory answer. When you are looking for your keys, but fail to notice them right in front of you. When you remember something incorrectly. When you try to tap a finger, but find yourself tapping the wrong one, all of these could be called system errors.
Because the brain does not run stable processes like a PC, it is difficult for me to continue this analogy. Like the rest of the body, cognition is a mash up of competing processes. When one fails, its role is absent, but this rarely causes a "halt" of any type, and generally goes unnoticed by the thinker except maybe in retrospect.
Broadly speaking, you have 3 major ways of communication, electrochemical/hormonal/and nutritional. The brain is aware and responsive to changes in all of these. Each part of the brain makes ado with different "hardware" designs, but sits in the same chemical bath. They are wired linearly, most primitive to most advanced, but the mid and fore brain as well as both lobes have direct trunks of communication to their various parts. There are nerves for receiving direct and indirect stimulus and a few areas that could truly be called neurological subsystems, such as the nerve half of the pituitary, and the omentum. Hormonal and Glial cell mediated performance changes are unparalleled in computing. Maybe if you thought of each brain as running several virtual machines, each geared for specific emphases in how it used the hardware but some even running redundant processes, and it could prioritize system resources towards each VM based on a completely different communication subsystem than that which it operated the bigger mechanical hardware or communicated to the other systems (other parts of the brain.) That secondary network can communicate to everything, both as a wifi like broadcast, and as a completely different wired network parasited onto the main one.
In short, the architecture is so different from computing as we know it, that I would suggest the premise of calling the brain a "biological supercomputer" is flawed.
The brain has so many fail safes and evolutionary carry overs that there is a lot that can "go wrong" that will not matter, from a continuation or clinical "quality of life" perspective, many things that may/do matter that we can still get by with, and then severe debilitating problems, that would still not equal death, and I can't say how any of them would correlate to "brain.exe" There is no such processes.
That said, we can easily overload various systems, but we are designed to adapt to that. The experience varies, depending on what system is overloaded. Could be confusion, foggy or clumsy thinking, missing the obvious, depression, headache, palpitation, nausea, pain, forceful exhaustion, delusion, hallucination. BUT these are all within the human condition, and quite normal experiences. Studying too hard, getting in a heated argument, staying up late, under or overstimulation of the senses, any of these can do it, but none are detrimental events. That is how capable we are.
One commenter mentioned babies being overloaded. Well, consider the student in lecture, when their eyes glaze over (as yours may have by now.) Their brains have checked out, a system is overloaded and other systems are taking priority, it is inevitable, but not detrimental, except to learning the class material via lecture. They are just more graceful and less hyper focused than babies, whom really do just rest their brain for a bit by zoning out, and then go back to their aggressive learning behaviors. Is it fair to call this "crashing?" What about when the same studnet, exhausted from the day, "crashes" on to the couch. It isn't like their brain stops.. even if they do, "check out" rather they just use other, less fatigued parts, maybe watch TV in a hypnotic daze, or browse the internet. Maybe they will be short tempered, maybe the will be suicidally depressed, maybe they will just be hungry, but all of these are particular systems demonstrating fatigue and entering low/alternately functioning states. The same is true of the coma victim.