Transcription is a vital stage in the process of gene expression and a major contributor to fluctuations in gene expression levels for which it is typically modeled as a single-step process with Poisson statistics. However, recent single molecule experiments raise questions about the validity of such a simple single-step picture. We present a molecular multistep model of transcription elongation that demonstrates that transcription times are in general non-Poisson-distributed. In particular, we model transcriptional pauses due to backtracking of the RNA polymerase as a first passage process. By including such pauses, we obtain a broad, heavy-tailed distribution of transcription elongation times, which can be significantly longer than would be otherwise. When transcriptional pauses result in long transcription times, we demonstrate that this naturally leads to bursts of mRNA production and non-Poisson statistics of mRNA levels. These results suggest that transcriptional pauses may be a significant contributor to the variability in transcription rates with direct implications for noise in cellular processes as well as variability between cells.