protein-binding sRNAs
The sRNA mediated regulation in bacteria operates via diverse mechanisms. This case of sRNA competing with a mRNA for a protein is a passive kind of regulation. This might be good for finetuning but may not be very efficient. It is much better to actively regulate a mRNA by direct binding. Also, it will work only if the concerned protein is in limiting concentrations.

I haven't come across any examples of what you are interested in. There may be one similar situation in case of a eukaryotic system. miRNAs can be titrated away by long RNAs that bear multiple binding sites for that miRNA, thus making the miRNAs unavailable for binding to the target mRNA. These long RNAs are called ceRNA (competing endogenous RNA) or miRNA-sponges (this term is mostly used if the RNA is exogenous). However this is not really a protein-RNA interaction even though RISC complex contains proteins.