Where the bound does not apply
The maximum M of a uniform(0,\theta) sample has variance \theta^2 n/\{(n+1)^2(n+2)\}, which for large n is about \theta^2/n^2. The Cramér–Rao calculation, if the regularity conditions are ignored and one pretends \mathcal{I}(\theta) is of order n, suggests a lower bound of order 1/n.
Explain which regularity condition fails, and why a variance of order 1/n^2 is not a contradiction.
The uniform density on (0,\theta) is zero for x>\theta, so the set where the density is positive depends on \theta. Differentiating under the integral is not legitimate, and the proof that the score has mean zero does not apply. A variance of order 1/n^2 is smaller than a formal bound of order 1/n only if that bound was valid. Here it was not. The maximum is a consistent estimator. It is not a counter-example to an inequality whose hypotheses fail.