In this paper, we introduce the concepts of Cesàro summability and Tauberian theorem in neutrosophic normed spaces. We study that Cesàro summability in neutrosophic norm space does not imply ordinary convergence, and we give an example in support of our statement. We define slowly oscillating sequences in neutrosophic normed spaces and prove that Cesàro summability of slowly oscillating sequences implies ordinary convergence in neutrosophic normed spaces. Finally, we define q−bounded sequence with respect to the neutrosophic norm and also show how it relates to oscillating sequence in neutrosophic normed spaces.