From practical experience, I find plotting the results of discretization refinement studies best to figure out whether or not the observed order of accuracy is "close enough". However, writing an automated test of a method's order of accuracy requires some sort of regression to calculate the observed order of accuracy from data, and then defining an appropriate $\Delta$ -- a comparison tolerance -- such that if the observed order of accuracy is within $\Delta$ of the theoretical order of accuracy, the test "passes".
I don't know of any theory that suggests an appropriate $\Delta$ or an appropriate way to bracket the asymptotic order of convergence region a priori. I have always written these kinds of tests by reverse-engineering them from plots, so I typically choose $\Delta = .2$ or $\Delta = .3$, because I see the order of accuracy deviate by at most that amount.
In certain cases (e.g., discontinuous initial conditions, or stiff problems), theory can predict how the order of accuracy of a method will degrade; if this additional theory isn't taken into account, then you might need to choose a larger $\Delta$ to compensate. For instance, in Section 8.7 of "Finite-Volume Methods for Hyperbolic Problems", Randy LeVeque discusses how convergence theory for a first-order upwind method assumes smooth solutions. If instead, a weak solution is discontinuous, the order of accuracy in space (in the $L^{1}$ norm) is one-half instead of one. So in a case like that, if you assume that the true order should be one, you might need to make $\Delta$ as large as 0.6, but in some cases, the degradation of the error is known, and you can adjust the theoretical order of accuracy accordingly.