I don't know where I heard this, and I can rephrase the question to make it more clear: Before a person begins practicing a martial skill, he or she has an instinctual reflex for defense, regardless of how ineffective it might be. Once training has begun, he begins to learn the proper actions, but it requires thinking. And that takes time. Because the response time is slowed by thought replacing reflex, the individual is actually more susceptible to offensive measures. That is, until which time, usually 1-3 years, the right actions become reflex, and there is no more conscious thought about the best reaction; in which case, the person is much more protected. I believe I heard this in relation to Aikido in particular. Is this true?
This is generally not true - there are many defensive arts where you improve your fighting skills right away and reach a basic proficiency within a matter of a few weeks or months at most (skill wise, at least, fitness can take longer to produce).
Many weapon based arts that are close to their original use also have this same thing - if you were going to go to war or be raided, getting you up to basic proficiency quickly was crucial. Having people get WORSE for 1-3 years was not something people could afford to risk.
A key part of combative training is moving past freezing up and overthinking things early in your practice. You might only learn a small handful of basic moves, but you drill them, you learn to do them faster, against resisting, live opponents, and you are often encouraged to adapt within the safety limits of the training exercises.
You may have periods where you feel worse or losing skill because you're restructuring your movement patterns, but it's not going to be worse than when you never trained at all, especially when you look at how poorly untrained people tend deliver strikes or how open they leave themselves when they go to grapple.
Internal Arts, Sure
I've heard of skilled folks who do arts that play with lots of subtle unbalancing, sensitivity, and sticking arts (like Tai Chi, and I imagine Aikido as well) talk about having a few years of being unable to do much with it until they develop the ability to sense and react in the movement methodology of their arts.
I haven't personally explored these arts deeply, so but that sounds parallel to the experience I often hear from these folks.
And then the liars
There's also the folks who have never really had to combatively use their art, never train to do so, and may have been taught in the same way from their own teachers. They talk about taking years and years to gain "basic ability" because... well, it's a way to excuse never having any development ever.
I think there is a 'superman' complex that many beginners suffer from. A notion that they are doing well in class maybe won a competition or two. Get to a real fight and the natural instinct to run our be aggressive is lost to trying to figure out whether to throw a punch or a kick.
This, along with semi or light contact training can give a false sense of security as well as poor distancing (for semi or light contact you are intentionally an inch or two away from where you would need to be to be effective, this can't easily be rectified in the moment of a real fight ).
Geoff Thompson talks of this in many of his books.
This is not a question of whether or not you have any martial martial arts training. The issue is self perception and subconscious signals. If you are uncertain about yourself and feel insecure, you may be thinking "oh no I'm going to be mugged" while walking about late at night. This self-hypnosis creates an air of uncertainty and insecurity, which potential attackers or muggers will pick up on, using subconscious signals, or "intuition".
I have had students telling of situations they've found themselves in, where they, in the aftermath wondered why they weren't raped or mugged, and my theory (I have read theories to back this up, I just cannot recall them off the top of my head) is that the martial arts training has given the practitioner a sense of calm and self confidence, which again may not be conscious, but gives off vibes that are in opposition to the previously mentioned.
When that is said, anybody can have a bad day. And if you go looking for trouble, you will find it (and street fighting does not follow the rules if the dojo).
If you refer only if the reaction time of instinctive reflexes being slowed by starting to train a martial art because you are thinking what to do, the answer is no.
Instinctive reflexes like crouching or closing the eyes when going to be hit, are really hard to modify. It doesn't take a few days not even months to modify them, and will be automatically triggered when the brain encounters an impending danger which to it cannot react in a toughtful manner quickly enough.
Reflexes are not a magical sense of danger nor spidey-sense like abilities, they are the lower natural automatic defenses and they are innate, as they are imprinted in neuronal networks from birth, if a person is aware of impending danger, it'll automatically cover, normally crouching or covering its face. But if the person is not aware of the danger it will not automatically cover because of its trainning, there is no such magic. But awareness can be increased by trainnning it as well, and in agree with Bankuei a little training in combat can make a great difference at the time of ACTUAL fighting.