Next-generation amplicon sequencing of 16S ribosomal RNA is widely used to survey microbial communities. Alpha and beta diversities of these communities are often quantified on the basis of OTU frequencies in the reads. Read abundances are biased by factors including 16S copy number and PCR primer mismatches which can cause the read abundance distribution to diverge substantially from the species abundance distribution. Using mock community tests with species abundances determined independently by shotgun sequencing, I find that 16S amplicon read frequencies have no meaningful correlation with species frequencies (Pearson coefficient r close to zero). In addition, I show that that the Jaccard distance between the abundance distributions for reads of replicate samples, which ideally would be zero, is typically ~0.15 with values up to 0.71 for replicates sequenced in different runs. Using simulated communities, I estimate that the average rank of a dominant species in the reads is 3. I describe UNBIAS, a method that attempts to correct for abundance bias due to gene copy number and primer mismatches. I show that UNBIAS can achieve informative, but still poor, correlations (r ~0.6) between estimated and true abundances in the idealized case of mock samples where species are well known. However, r falls to ~0.4 when the closest reference species have 97% identity and to ~0.2 at 95% identity. This degradation is mostly explained by the increased difficulty in predicting 16S copy number when OTUs have lower similarity with the reference database, as will typically be the case in practice. 16S abundance bias therefore remains an unsolved problem, calling into question the naive use of alpha and beta diversity metrics based on frequency distributions.