Only one Cart Discount can be applied at a time. A manually-entered code will take precedence over an auto-applied "current promotion" discount.
However, Item Discount codes (defined in each product's settings as a flat amount off the product's unit price) can be combined with a Cart Discount, so you might consider setting your volume discount as an Item Discount for the product(s) and tell people they can enter that code when they order 10 items or more. If you have multiple different products that qualify for your volume discount, you can give them each the same Item Discount code, and if their prices differ, you can even give each product a different discount amount off (so it works out to the same percentage in each case, since Item Discounts cannot be configured as a percentage). Entering an Item Discount code in the cart would apply the discount to all products which have a match for that code in their settings.