Network meta-analysis has played an important role in evidence-based medicine for assessing the comparative effectiveness of multiple available treatments. The prediction interval has been one of the standard outputs in recent network meta-analysis as an effective measure that enables simultaneous assessment of uncertainties in treatment effects and heterogeneity among studies. To construct the prediction interval, a large-sample approximating method based on the t-distribution has generally been applied in practice; however, recent studies have shown that similar t-approximation methods for conventional pairwise meta-analyses can substantially underestimate the uncertainty under realistic situations. In this article, we performed simulation studies to assess the validity of the current standard method for network meta-analysis, and we show that its validity can also be violated under realistic situations. To address the invalidity issue, we developed two new methods to construct more accurate prediction intervals through bootstrap and Kenward-Roger-type adjustment. In simulation experiments, the two proposed methods exhibited better coverage performance and generally provided wider prediction intervals than the ordinary t-approximation. We also developed an R package, PINMA (https://cran.r-project.org/web/packages/PINMA/), to perform the proposed methods using simple commands. We illustrate the effectiveness of the proposed methods through applications to two real network meta-analyses.