Hi Chadmc,
As you are measuring the ultimate tensile stress which depends on area and thus the diameter, so I think, if you are testing then cutting the same sample to test again probability is that it is due to strain Hardening, but also what I also think is, that the load sustaining capacity of the material does not change as the material does not change as all the samples are separate,then due to the reduction in the area, the stress is increasing.
To the point that it is not increasing proportionately, I really don't have a good point although I think it is due to the removal of material.
This is just my opinion to the problem, Kindly enlighten me if I have gone wrong anywhere as I am still learning.