Based on a maximal inequality-type result of Cuculescu, we establish some noncommutative maximal inequalities such as the Hajék–Penyi and Etemadi inequalities. In addition, we present a noncommutative Kolmogorov-type inequality by showing that if $x_1,x_2,\dots ,x_n$ are successively independent self-adjoint random variables in a noncommutative probability space $(\mathfrak{M},\tau )$ such that $\tau \left(x_k\right)=0$ and $s_k{s}_{k-1}=s_{k-1}{s}_k$, where $s_k={\sum }_{j=1}^k{x}_j$, then, for any $\lambda >0$, there exists a projection $e$ such that

$$1-\frac{(\lambda +{max\hspace{0.17em}}_{1\le k\le n}\Vert x_k\Vert {)}^2}{{\sum }_{k=1}^{n}var\left(x_k\right)}\le \tau (e)\le \frac{\tau \left(s_n^2\right)}{{\lambda }^2}.$$ As a result, we investigate the relation between the convergence of a series of independent random variables and the corresponding series of their variances.