For an integer sequence (with even sum), the closer that the sequence is to being regular, the more likely that the sequence is graphic. But how regular must a sequence be before it must always be graphic? We show that for many sequences if all values are within $\frac{n-2}{4}$ of the mean degree value, then the sequence is graphic. We also see how this result extends to show when a maximum difference between sequence values implies that a sequence is graphic.