Nucleic-acid-binding proteins are generally viewed as either specific or nonspecific, depending on characteristics of their binding sites in DNA or RNA. Most studies have focused on specific proteins, which identify cognate sites by binding with highest affinities to regions with defined signatures in sequence, structure or both. Proteins that bind to sites devoid of defined sequence or structure signatures are considered nonspecific. Substrate binding by these proteins is poorly understood, and it is not known to what extent seemingly nonspecific proteins discriminate between different binding sites, aside from those sequestered by nucleic acid structures. Here we systematically examine substrate binding by the apparently nonspecific RNA-binding protein C5, and find clear discrimination between different binding site variants. C5 is the protein subunit of the transfer RNA processing ribonucleoprotein enzyme RNase P from Escherichia coli. The protein binds 5' leaders of precursor tRNAs at a site without sequence or structure signatures. We measure functional binding of C5 to all possible sequence variants in its substrate binding site, using a high-throughput sequencing kinetics approach (HITS-KIN) that simultaneously follows processing of thousands of RNA species. C5 binds different substrate variants with affinities varying by orders of magnitude. The distribution of functional affinities of C5 for all substrate variants resembles affinity distributions of highly specific nucleic acid binding proteins. Unlike these specific proteins, C5 does not bind its physiological RNA targets with the highest affinity, but with affinities near the median of the distribution, a region that is not associated with a sequence signature. We delineate defined rules governing substrate recognition by C5, which reveal specificity that is hidden in cellular substrates for RNase P. Our findings suggest that apparently nonspecific and specific RNA-binding modes may not differ fundamentally, but represent distinct parts of common affinity distributions.