Attaching Unique Molecular Identifiers (UMI) to RNA molecules in the first step of sequencing library preparation establishes a distinct identity for each input molecule. This makes it possible to eliminate the effects of PCR amplification bias, which is particularly important where many PCR cycles are required, for example, in single cell studies. After PCR, molecules sharing a UMI are assumed to be derived from the same input molecule. In our single cell RNA-Seq studies of Physcomitrella patens, we discovered that reads sharing a UMI, and therefore presumed to be derived from the same mRNA molecule, frequently map to different, but closely spaced locations. This behaviour occurs in all such libraries that we have produced, and in multiple other UMI-containing RNA-Seq data sets in the public domain. This apparent paradox, that reads of identical origin map to distinct genomic coordinates may be partially explained by PCR stutter, which is often seen in low-entropy templates and those containing simple tandem repeats. In the absence of UMI this artefact is undetectable. We show that the common assumption that sequence reads having different mapping coordinates are derived from different starting molecules does not hold. Unless taken into account, this artefact is likely to result in over-estimation of certain transcript abundances, depending on the counting method employed.