Thank you for your input.
My RNAseq data derives from 125 bp paired-end reads.
I should have stated more clearly that the "errant" bases are at the 3' end of the reads, not the 5'. Apologies for not including an image earlier:
I believe I did a good job of removing adapter sequences as nothing comes up on the "overrepresented sequences" report. Since the problem I'm concerned with is at the 3' end of paired-end reads, it isn't clear that biased fragmentation could account for the observation (although that could explain the observations at the 5' end of the reads in the image above).
So, does this new information give you any more ideas or have I missed something? Thanks again.