I'll just write briefly where this question comes from. Given two positive measures $\mu, \nu$ in $\mathbb{R}^n$, we call the upper/lower densities of $\nu$ with respect to $\mu$ the quatities:
$$ D_\mu ^+ \nu (x) = \limsup_{\rho \to 0} \frac{\nu(B_\rho (x))}{\mu (B_\rho (x))} \text{ and } D_\mu ^- \nu (x) = \liminf_{\rho \to 0} \frac{\nu(B_\rho (x))}{\mu (B_\rho (x))}.$$
Instead, we define the upper/lower k-dimensional densities of $\mu$ as:
$$ \Theta ^+ (\mu ,x) = \limsup_{\rho \to 0} \frac{\mu(B_\rho (x))}{\omega_k \rho^k} \text{ and } \Theta^- (\mu ,x) = \liminf_{\rho \to 0} \frac{\mu(B_\rho (x))}{\omega_k \rho^k}.$$
Now there are results for both of these quantities, but they are different. For example, there are some estimates on the lower density that do not pass to the lower k-density. This should mean that there are no measures $\nu$ in $\mathbb{R}^n$ such that $\nu(B_\rho (x))= \omega_k \rho^k$, otherwise the second definitions would simply be a particular case of the first ones. But why doesn't such a measure exist?
Ok I was just reviewing my notes, and thinking about it it's simple (at least intuitively, I'm not writing a full proof). Consider the case of a circle $C$ in the plane of radius 1. If you consider two smaller circles $C_1, C_2$ of radii 1/2 centered in a diameter so they fit inside the first circle and are disjoint, then by monotonicity:
$$C_1 \cup C_2 \subset C , C_1 \cap C_2 = \emptyset \implies \mu (C_1) + \mu(C_2) \leq \mu(C) .$$
But if $\mu$ had to behave like the $\mathcal{L}^1$ measure of the diameter of those circles, both terms are equal to 1. Put inside some other disjoint circles and get an absurd.
I haven't done a rigorous proof in full generality, but I bet you can always find a counterexample like this.