Is there anything worse than an overcooked vegetable? Doubt it. What vegetables did your parents make you eat even though they were overcooked and disgusting?
Asparagus. Such a fantastic veggie. So terrible when overcooked. It gets slimy and stringy.
String beans. I nearly came to blows with my dad (who made our dinners) when I was ten about how to make strings beans taste good. Lightly steam (you really should blanch, shock, and saute them) and then serve. No boiling or over-steaming.
Brussels Sprouts. Never boil. Always roast. Eating boiled Brussels Sprout is the same as licking a dirty toilet. Blanche. Saute. Roast. Blanche. Do anything else than boil.