We consider the estimation of an integer vector $\hbx\in \mathbb{Z}^n$ from the linear observation $\y=\A\hbx+\v$, where $\A\in\mathbb{R}^{m\times n}$ is a random matrix with independent and identically distributed (i.i.d.) standard Gaussian $\mathcal{N}(0,1)$ entries, and $\v\in \mathbb{R}^m$ is a noise vector with i.i.d. $\mathcal{N}(0,σ^2 )$ entries with given $σ$. In digital communications, $\hbx$ is typically uniformly distributed over an $n$-dimensional box $\mathcal{B}$. For this estimation problem, successive interference cancellation (SIC) decoders are popular due to their low complexity, and a detailed analysis of their word error rates (WERs) is highly useful. In this paper, we derive closed-form WER expressions for two cases: (1) $\hbx\in \mathbb{Z}^n$ is fixed and (2) $\hbx$ is uniformly distributed over $\mathcal{B}$. We also investigate some of their properties in detail and show that they agree closely with simulated word error probabilities.