I agree with HalfWavy -- cutting your hair short seems like overreacting!
Look at it this way. Say your hair grows half an inch per month -- that's considered average. Bra-strap-length hair is probably about 20 inches long at least. That would mean your longest hairs are almost 4 years old. For you, that may be how long your hairs last before naturally falling out. Your hair is still growing, but you don't see an increase in length.
Keeping your hair healthy as HalfWavy suggested would be a better approach, I think.