The problems begin when training is closer in appearance to fighting. For boxing the most confusing situation would likely be a speed bag workout. Yet we know that boxers don't hit with the side of their hands, with the fists making a circular motion in front of his face. The boxer is training timing and hand eye coordination. The speed bag conditions him to see an object moving very quickly towards his face and hit it at just the right time. It also helps condition the arms by holding them high and in motion for a long time. But few people would confuse that training with actual fighting. It's one aspect of a simulated fight. The training isolates one or a few skills the fighter needs, and allows him to work on them without worrying about all the other factors of an actual fight.
An observer not versed in karate may be prone to see basics (kihon) or solo forms (kata) and be quick to say the training is wrong or ineffective, because no one fights like that. The fixed positions in low stances, often with one hand at the hip, does not look like effective fighting posture. The observer who stays for sparring practice would see something that on its surface looks completely different, with advanced practitioners keeping their hands in front, making circular and continuous movements, maybe even bouncing on the balls of their feet in high stances like a boxer.
But the kata is training, not fighting. The low stances make for powerful legs. The hand at the hip encourages contraction of back and abdominal muscles when delivering a technique with the opposite hand. The dramatic turns encourage development of rapid hip rotation and balance, providing the control needed to sidestep an attack during fighting.
The worst arguments come between related styles, often from those who have not been exposed to multiple styles, or those who on some level, worry that the study in which they've invested so much effort may not be the magic bullet they expected. Styles of practice become dogma that must be defended against heresy, and argued for like politics or religion.
Chinese martial arts typically place a high value on continuous, relaxed motion. Karate places high value on finishing all techniques with focused power. Yet advanced kung fu or wushu training will certainly include focused punching drills or work hitting objects such as padded bags, and karate has some kata that have more circular and continuous motions than the basic forms.
One can easily find heated arguments on minor technical points, such as the final position of the fist when punching. Worst are when practitioners start making recourse to physics or body mechanics, often when those same proponents have not taken a college level physics course, or don't have a degree in sports medicine. The most fierce arguments will often be those in any field where there is not one clear answer. There may be questions of what is believed to be the most important factor among many competing issues. A fully rotated karate fist may appear to have more snap or focus. It also may place much more stress on the wrist when actually brought into contact with the target. Maybe it takes a fraction longer to deliver. It will not be the right weapon for every target. What aspect is most important? That answer is that it depends on the situation, and on the training method.
However, one should not discard all rules of training as mere dogma. By adopting certain rules in training, if considered as part of a consistent style, the student is getting a complete education within a given framework. The student who flits from style to style does not get the benefit of a curriculum that teaches different aspects at different times, according to the level of the student.
Tai Chi is an style that looks little like fighting, and in the west has come to be practiced most often by older people, who have no interest in fighting, and are often at a point in their lives when physical limitations render combat training nearly impossible. But tai chi was not historically the style just of the old, but often taught in conjunction with kung fu styles that emphasize rapid and more athletic motions. Tai chi is more basic and focused training in relaxation, body mechanics, and, in push hands training, leverage against an opponent. All these elements can be taught in different ways at different times. An intermediate tai chi student will probably be more relaxed, with better balance than a karate student with a comparable duration of training. However, he will lack the speed and power of the karateka. With advanced training, both students should arrive to a similar level of capability, although likely with some lasting different emphasis or preferences.
Consistent with their training priorities, tai chi styles typically insist on turning a foot only when unweighted. Stability of stance and relaxation is given priority. In contrast, karate students are taught not to turn an unweighted foot because it telegraphs the movement to the opponent. Which is correct? The answer is, it depends. A tai chi move in combat would more typically be close in fighting, drawing the opponent into an unbalanced position where the tai chi practitioner can apply a joint lock or throw. Telegraphing the move is less important since the fighters will already be in contact. A karate practitioner keeping his opponent at distance, waiting for one powerful strike, certainly must not telegraph the move. Which should the student practice? Both styles and practitioners start to look more similar at advanced levels. A wise student will stick with one style throught his life, but gain exposure to other styles after black belt level, or 10 years of study in his core style.
Personally, I've found that a friend of mine (named Joel), who has studied many different styles while centering his studies on Shotokan, has some of the most helpful advice with regards to training in many styles where basics may conflict. His advice was to treat each approach or technique as an opportunity for mastery over your own body. If one technique feels awkward, or conflicts with the teaching you believe is correct, you have the opportunity to treat it as just another exercise to master, because your body should be able to do whatever you tell it to. In that way it is easy to separate helpful aspects of instruction that you may benefit from, without become overly concerned with the aspects that you don't.