Summary: Let \(H = - \Delta + | x |^2\) be the Hermite operator in \(\mathbb{R}^n\). In this paper we study almost everywhere convergence of the Bochner-Riesz means associated with \(H\) which is defined by \(S_R^\lambda(H) f(x) = \sum_{k = 0}^\infty ( 1 - \frac{ 2 k + n}{ R^2} )_+^\lambda P_k f(x)\). Here \(P_k f\) is the \(k\)-th Hermite spectral projection operator. For \(2 \leq p < \infty \), we prove that \[ \lim_{R \to \infty} S_R^\lambda(H) f = f \text{ a.e. }\] for all \(f \in L^p( \mathbb{R}^n)\) provided that \(\lambda > \lambda(p) / 2\) and \(\lambda(p) = \max \{n(1 / 2 - 1 / p) - 1 / 2, 0 \} \). Conversely, we also show the convergence generally fails if \(\lambda < \lambda(p) / 2\) in the sense that there is an \(f \in L^p( \mathbb{R}^n)\) for \(2 n /(n - 1) \leq p\) such that the convergence fails. This is in surprising contrast with a.e. convergence of the classical Bochner-Riesz means for the Laplacian. For \(n \geq 2\) and \(p \geq 2\) our result tells that the critical summability index for a.e. convergence for \(S_R^\lambda(H)\) is as small as only the half of the critical index for a.e. convergence of the classical Bochner-Riesz means. When \(n = 1\), we show a.e. convergence holds for \(f \in L^p(\mathbb{R})\) with \(p \geq 2\) whenever \(\lambda > 0\). Compared with the classical result due to R. Askey and S. Wainger [Am. J. Math. 87, 695–708 (1965; Zbl 0125.31301)] who showed the optimal \(L^p\) convergence for \(S_R^\lambda(H)\) on \(\mathbb{R}\) we only need smaller summability index for a.e. convergence.