No

I think this confusion comes about from the idea of 'net carbs'. It's not that the fiber in, say, an apple in any way cancels out the carbs in the apple, but rather that some of the carbs in the apple are in fact fiber.

Insoluble fiber is primarily cellulose, which plants use as structural material in their cells. This is the primary component of wood, fruit and vegetable cell walls, and paper. It is non digestible (by humans) and helps to bulk up stool, which is why it helps keep one regular.

Cellulose is a carbohydrate, a sugar, in fact, and thus it adds to the carb number on the nutritional data of a food, even though it is undigestable. However, it is also logged as fiber on the data, so to find the digestible carbs (net carbs) in an apple, you can subtract the fiber number from the total carbs number and get something pretty close.

So the amount of fiber you eat doesn't cancel out sugar or starch carbs. The fiber itself is a carb, but since you pass it through undigested, it doesn't contribute to the total carbs your body has to burn.