megan,
Is short, no. It's a design requirement that full elevator in available up till Vmo without overstressing the airframe, and that would not be sufficient to generate positive lift on the tail.
BTW, it isn't the negative lift that is needed for stability- just that the lift component of the tail changes sufficiently with changes of AofA to ensure a corrective moment, while not stalling.
It is the last bit that means that most tails create negative lift, but it is not an actual requirement.