How much toothpaste has to be swallowed to cause a toothpaste overdose?

Depends: It depends on toothpaste Fluoride concentration. To prevent it, children must be supervised until healthy toothbrushing habits have developed.

Child < adult: For an adult, a family size full tube of Fluoride toothpaste, since Fluoride can cause tummy upset (vomiting/diarrhea). Non-fluoride toothpaste will not cause as much of a problem. For children, about half a tube of Fluoride toothpaste at one time. Once again, tummy upset/pain. At that point, calling the pediatrician is the best advice.

Probably: Probably about a bathtub full at one time. The one exception is fluoride containing toothpastes. Fluoride is a wonderful public health measure that dramatically reduces incidence of cavities, but must be used in measured dose. For example, children under age three should only use s small smear of toothpaste on their brush, over age three a pea-sized portion. Overuse can cause tooth mottling.