FrogReaver
The most respectful and polite poster ever
It wouldn't be such a bad thing, since you wouldn't need to worry about interactions between class abilities, but it would still be a bad thing. As you add more classes, each existing class becomes less distinct, and it's easy to dilute class identity so far down that you can't even tell what class someone is by looking at them.
If you add a samurai class and a knight class, then the fighter class becomes less clear in what it is, and it becomes much more difficult to tell which of those three classes is the correct way to represent the non-magical warrior-type with the plate armor and the big sword.
To me the character name doesn't have a direct correlation to the game world. A fighter class can represent anyone from a noble night, a holy warrior, a thug, a nomadic warrior, a soldier, a sailor, etc. All of those can be represented in game as a fighter. Some can also be represented as a Paladin or a rogue or a ranger or a barbarian etc.
I'm not of the opinion that classes cannot overlap in concept. When they do then you just have more choices on how to arrive at your concept.