It depends on what kind of plant and what your plans are for it.
If it is a monocot (palm, grass, philodendron) it doesn’t matter how “root bound” it gets. The roots don’t get any thicker than when they first grow.
If it is a dicot (ficus, oak, holly, rose) the roots get thicker as time goes by and they can girdle the stem. This is especially important for trees that are going to be planted in the ground. It can cause poor growth and even death when the roots cut off nutrient and water flow to the rest of the tree.
A special case is plants that will never go into the ground, like houseplants. They can be “bonsaied” by removing tangled roots and depleted soil. These don’t even need to go into a bigger pot.
If it is a tree or large shrub it CAN go into a much bigger pot as long as watering is careful to prevent sogginess at depth. This way you can prevent curling roots by giving them more room.