The problem isn't really overlap of technique, it's that different arts have different ways of approaching things on a fundamental level (below technique), and to have conflicting approaches can be highly counter-productive when it comes to these arts.
I'll see if I can explain what I mean.
First off, we'll look at what makes a martial art in the first place (I should just have this on file somewhere!). But before we even get to that, I'll point out a few things that a martial art is not. A martial art is not it's techniques. They are an expression of what it really is. A martial art is not something you can consciously call on when needed. It is not trained that way, and it shouldn't be. A martial art is not for self defence. That is not it's aim, despite whatever the instructor advertises it to be. A martial art is not lacking when you really understand it. Limitations only exist in technique, and as I said, that is not what a martial art is.
So what is a martial art? Well, in simple terms, a martial art is a vehicle for passing on particular lessons and approaches designed to instill the particular philosophy that that art espouses. And those lessons are designed not to be instilled in the conscious (thinking) part of your mind, but in the unconscious (automated) part of yourself. And that will remove any belief that you can use X-art for this situation/range, and Y-art for another. The unconscious just doesn't work that way.
This philosophy (the guiding principles behind any martial art, traditional, modern, or sport), combined with the human element (who, where, when, why) is what gives you the techniques that most people think of when they think of a martial art. Essentially, to take an art like Aikido, the philosophy involves not forcing your way upon the world, instead blending with the energies around you. This lends itself to non-resisting actions, circular re-direction, and not causing unnecessary damage to another person. The human element involves Ueshiba Morihei, a rather short, lightly framed man (also of advanced years by the end of Aikido's development), who was Japanese. This lends itself to grappling techniques over striking, use of anothers energy rather than physical strength, lead hand/foot rather than rear hand for power etc. With the philosophy and human element combined within the field of human combative applications you the have Aikido.
So that's (in a very basic way) what a martial art actually is. But the other thing to realise is the way it is taught. Martial arts are taught by repeating certain actions (techniques) over and over again. The common thing you will hear is that this is to ingrain "muscle memory", and that's kinda true, but not really the important part. What you are really doing is imprinting a specific process, essentially the specific philosophy, and the way it is expressed (power source, angles, tension/relaxation methods, ways of moving etc) on your unconscious programming. By repeating it over and over you are telling your unconscious that "this is important", or "this is strong". If you are doing that with two martial arts at the same time, you are essentially telling your unconscious that both are important and both are strong. The problem is that the unconscious will always choose the best of any two options, and if you have opposing martial approaches, only one will come out. If you're lucky.
The worse-case scenario is that the unconscious will actually not think either are strong enough, as there must be some doubt for you to be training more than one, and as a result, nothing will come out. This is where freezing comes into it. Basically, your unconscious, under the stress of a real situation, looks to what it knows to be strong, powerful, and reliable, and if you have constantly undermined your belief in both systems by contradicting them with another one, neither will be considered strong, and there will be nothing there for you to find when you need it, no matter how much time you have trained them.
But there are ways around this. Essentially, it revolves around having one system that is "your" system, and additional knowledge is added to that from the second one. In this case you train for a number of years in a single system, ingraining the one method/philosophy. When you say things like "Karate is more offensive, Judo for defence", I would suggest that you don't quite get either art, as both are defensive and offensive, depending on application, you just need to apply them that way. As I said, a martial art is not limited, only the techniques are. By that I mean that a single martial art, following it's single philosophy, actually provides you with all the answers to any situation you may need to apply it. Karate doesn't really deal much with ground work, but the philosophies can certainly be applied to ground fighting provided you understand them and the environment of the ground well enough. So cross training here can help you understand a different environment, however when you are in a real situation you will still use the dominant system that your unconscious knows and believes is powerful, and that will most likely not be the ground fighting system you have cross trained in.
So the end result is simple. Until you have established a particular system, and gotten a real grounding in it, I highly recommend you don't try more than one system. Particularly if you have just started one. There are too many pitfalls, and really no benefits.
