Answer: What you describe sounds like the work of cutworms. These grublike critters live in the soil and emerge overnight, curl themselves around a tender plant stem and chew their way through. You can keep that from happening by placing paper collars around your new seedlings (a 2-inch length of the cardboard tube of a papertowel roll works well). You can cut it long ways, place it around your seedling, burying it a half inch or so in the soil so it will stay in place. Or, you can surround the stem with a few toothpicks. Anything that will keep the cutworms from curling around the stem will work. Once the seedlings are up and growing, cutworms shouldn't cause so much damage.
You can also use Bacillus thuringiensis (Bactur, Dipel, SOK-BT, Thuricide). Follow label directions and you should be able to rid your garden of cutworms.
Best wishes with your veggies!
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