To be concise, I am wondering whether there are natural examples of probability measures $\mu$ compactly supported on the real line which satisfy $\mu(I) \lesssim l_n^\alpha$ for all intervals $I$ with $|I| = l_n$, for some $0 \leq \alpha \leq 1$ and a certain sequence $l_1 \geq l_2 \geq \dots \to 0$, but for which the general inequality $\mu(I) \lesssim |I|^\alpha$ fails.

Often, in order to lower bound the Hausdorff dimension of a certain set $X$ by a constant $\alpha$, one applies Frostman's lemma, finding a natural probability measure $\mu$ supported on $X$ with $\mu(I) \lesssim |I|^\alpha$. Often, one constructs $X$ as a fractal, as a limit of sets $X_n$, each a union of intervals of length $l_n$ (i.e. $X_1 \supset X_2 \supset \dots \to X$). Often, one obtains $\mu(I) \lesssim l_n^\alpha$ for all intervals $I$ of length $l_n$, and then tries to 'cheat out' the general bound from these estimates. My question is whether there are natural counterexamples of 'fake' $\alpha$ dimensional probability measures for which one can obtain the bound $\mu(I) \lesssim l_n^\alpha$ for all intervals $I$ of length $l_n$, where $l_1 \geq l_2 \geq \dots$ is a sequence converging to zero, but for which the bound $\mu(I) \lesssim |I|^\alpha$ does not hold for *all* intervals.

faithfulnessnotion for Hausdorff dimension. You might want to look here, especially Theorem 1. $\endgroup$