Can you ever reduce your reaction time? As a martial arts practitioner, you can learn to spot certain tiny movements of your opponent ahead of their attack and start your countermove earlier, but your reaction time stays the same, doesn't it?
First off, as @mattm rightfully points out, you need to sleep well and enough, drink enough water and eat well, live an active life etc. All these factors are necessary conditions for good reaction times.
Reacting to visual stimuli: time frames
Here, what cannot be overcome is the need to process the visual data through pattern recognising routines and then send the signal through the motoric central in the brain and the Central Nervous System to the muscles.
Therefore, there is a lower limit of about 180ms even for the simplest possible visual and motor patterns. Since attacks in martial arts are much more complex visual patterns, as well as the motoric patterns that are proper reactions are, this can go well beyond 500 ms: there are studies 1 2 which suggest that even when reacting to a very simple visual stimulus you are waiting for with a very simple and predetermined move, we are already well beyond 300ms for professional boxers (320+ms for a fast jab, 500+ms for a powerful jab as a goal).
Now, how to train the reaction time?
Basically in three ways:
You train to see the tension and kinetic chain built up in your opponent's body by learning about all these little clues (visual pattern recognition training). That way, you can react before the technique is even coming at you.
You train the motor pattern (move) of the suitable reactions to become as fast as possible. That way, you both become faster in the execution and in the time it needs to build up the corresponding action potentials in your motor cortex (brain). This is often called "building muscle memory" these days.
You train the link between them by being exposed to the visual stimulus and doing the corresponding reaction pattern over and over. This way the link between them becomes stronger to the point where you don't have to think about it anymore (neurologically speaking, thinking is a reroute via the frontal cortex). Also, neural links are established.
We are speaking about time frames of months to years here though. There is no fast track.
Conclusion and TL;DR
While there are lower bounds to reaction times due to the way our brains work, we can certainly train reaction times. Skill training does establish neural structures that enable us to make use of trained reaction patterns much faster and without having to think about it.
You can train to reduce reaction time, but there are fundamental limits originating from the human visual system, for example. Untrained individuals generally have slow reaction times because they are trying to identify what is happening to them and make decisions. You can train to eliminate this delay, but you can't reduce human visual system reaction time.
There are also well-known ways to impair your reaction time such as drinking alcohol or sleep deprivation. If you want to reduce your reaction time, get enough sleep and avoid alcohol.
*Can I offer a simplified answer - the instinctive centre of the human body is where our survival perceptions and processes are sited. So any self-defence actions may be traced here; in contrast 'thinking' what to do, and emotions will nearly always make a person slower in perceptions and actions. So to centre the mind in the instinctive centre is optimal. How to learn to do this? Practice of the forms and skills of martial arts until they become second nature; and following the precepts of tai chi regarding 'quieting the mind' enable greater sensitivity to the bodies rhythmns and instincts. In a serious and threatening situation (you are hopefully not responsible for), the instinctive centre is way ahead of your conscious mind. It will perceive not only someone's actions, but even intentions towards you, perceptions which are felt in the stomach. As soon as this is felt you can act decisively however that may be. This ability to instinctively 'sense' should be worked towards in life daily, as 'where the mind is, energy is'... Always trust your instinct. If you feel someone is going to attack you it is probably in their mind. Sooo, if someone is acting aggressively towards you and you feel physical attack is intended, you may act in self-defence to stop it. If they start to take something out of their pocket, end it before it begins, nothing good can come of allowing someone to take a weapon out. Legally it is justifiable if warning signs have been displayed;
*creating an argument about nothing;
*negative emotions of any sort/ heightened emo. state;
*invading personal space. (put your arm out full-length, and say 'this is my personal space - do not enter it again';
*verbal threats of violence/ increase of obscene language;
*NVC indicators; voice tone,'muscular' posture, chest out,chin forward, etc;
*positioning, ie trapping you in a corner/blocking your exits;
*pointless discussion on their terms;
And so on. Stay relaxed and ready to act; don't get caught out by appearances, or anything else. Readiness and confident relaxation will ensure you have a head-start. Best of luck and enjoy the martial arts, they're amazing systems of betterment.