You may end up having to spend quite a few first aid kits on cleaning wounds and whatnot, yes. Also cauterizing bleeding wounds can cause a lot of pain, which can further effect your performance throughout the rest of the day. Some good traits to take in order to offset this are the tough trait, which will increase the maximum health of all your limbs by 20%, effectively meaning you can go longer without having to heal yourself, and the pain-resistant trait which will just make you a better fighter overall since you won’t feel the same negative effects from getting hit as someone without the trait would.
Common? Not really. Possible? Sure. Once you get your skills to a certain threshold though you should technically not have to worry too much in small fights, especially if you keep the tactics I mentioned above in mind and stay generally wary. Martial Arts styles can help with this as they can give you special attacks that can have other affects on your enemies, such as taekwondo which I believe can give you a stunning blow at unarmed level 4 (****). Also many martial arts give you bonuses to blocking and dodging which can further improve your ability to get out of skirmishes unscathed. I’ve even been able to block bear swipes on more than one occasion - completely negating all damage. Having high melee/dodge skills has saved my life countless times, I tell you.
Also it’s important to wear protective clothing if you can get it. The tailoring skill can help to reduce any skill penalties that more constrictive clothing might impart, and also to repair and reinforce clothing, making it more protective overall. As you get better at fighting, you’ll be able to afford to wear heavier armour as well as you’ll be able to take the penalties and still fight effectively. Leather pants and jackets, army pants, and kevlar vests are all good armour, but all impose penalties that can’t be reduced even with tailoring (with the exception of the army pants, I believe). I always wait until I have at least 4 points in melee/dodge before I try wading into melee combat wearing that level of armour.