Can you put too much insulation in a wall?

Yes, you can put too much insulation in a wall in the sense of compressing insulation into a cavity that is too small for it, which actually reduces performance and can cause problems, although you cannot have too much R-value if installed correctly with proper space. For example, stuffing an R-30 batt designed for 9.5 inches into a 3.5 inch 2x4 wall does not give you R-30, it gives you about R-12 to R-13 because compression eliminates air pockets that provide insulation, and it can bow drywall, create gaps, and trap moisture. Similarly, over-packing dense-pack cellulose beyond its design density of 3.5 pounds per cubic foot can cause excessive pressure on sheathing and drywall. In terms of moisture, adding interior vapor barriers plus too much interior insulation in a cold climate without exterior insulation can cause the sheathing to stay cold and accumulate moisture, leading to rot. From a cost perspective, beyond about R-20 to R-23 in a 2x6 wall cavity, adding more cavity insulation has diminishing returns and it is better to add continuous exterior foam. So you cannot have too much R-value in principle, but you can have improper installation by overstuffing, which is counterproductive.