Summary: In this article, we try to give an answer to the simple question: “What is the optimal growth rate of the dimension \(p\) as a function of the sample size \(n\) for which the Central Limit Theorem (CLT) holds uniformly over the collection of \(p\)-dimensional hyper-rectangles ?”. Specifically, we are interested in the normal approximation of suitably scaled versions of the sum \(\sum_{i=1}^nX_i\) in \(\mathcal{R}^p\) uniformly over the class of hyper-rectangles \(\mathcal{A}^{re}=\{\prod_{j=1}^p[a_j,b_j]\cap \mathcal{R}:-\infty \leq a_j\leq b_j \leq \infty, j=1,\ldots ,p\} \), where \(X_1,\dots ,X_n\) are independent \(p\)-dimensional random vectors with each having independent and identically distributed (iid) components. We investigate the optimal cut-off rate of \(\log p\) below which the uniform CLT holds and above which it fails. According to some recent results of V. Chernozhukov et al. [Ann. Probab. 45, No. 4, 2309–2352 (2017; Zbl 1377.60040)], it is well known that the CLT holds uniformly over \(\mathcal{A}^{re}\) if \(\log p=o\left (n^{1/7}\right)\). They also conjectured that for CLT to hold uniformly over \(\mathcal{A}^{re} \), the optimal rate is \(\log p = o\left (n^{1/3}\right )\). We show instead that under some suitable conditions on the even moments and under vanishing odd moments, the CLT holds uniformly over \(\mathcal{A}^{re} \), when \(\log p=o\left (n^{1/2}\right)\). More precisely, we show that if \(\log p =\epsilon \sqrt{n}\) for some sufficiently small \(\epsilon >0\), the normal approximation is valid with an error \(\epsilon \), uniformly over \(\mathcal{A}^{re} \). Further, we show by an example that the uniform CLT over \(\mathcal{A}^{re}\) fails if \(\limsup_{ n\rightarrow \infty } n^{-(1/2+\delta )} \log p >0\) for some \(\delta >0\). Therefore, with some moment conditions the optimal rate of the growth of \(p\) for the validity of the CLT is given by \(\log p=o\left (n^{1/2}\right )\).