What is the reason for having an extra recognition site for a restriction enzyme?
The scenario in your question can be explained if the "extra" recognition site mapped very close to another recognition site for the same enzyme.

For example, if there were 2 EcoRI sites separated by between 0-50 bp of plasmid DNA, then depending on the type of gel being used, the resulting tiny DNA fragment generated could be difficult to detect (either because small fragments run off the bottom of the gel, or because they don't resolve well unless the gel contains a higher percentage of agarose or acrylamide). 

Similarly, depending on the size of the adjacent fragments in the various digests, and this assay's limitations on estimating fragment sizes, you might not detect that an "extra" fragment was generated, and "missing".