Does the sample maximum pin down a uniform ceiling?

Observations $X_1, \dots, X_n$ are i.i.d. Uniform$(0, \theta)$ with unknown ceiling $\theta$. You estimate $\theta$ by the largest value seen, $\hat\theta = X_{(n)} = \max_i X_i$, which is always strictly below $\theta$.

Is $\hat\theta$ a consistent estimator of $\theta$, despite always underestimating it?