Very strange. In the demographics, the part of the name which appears to show up are in red above it and bold black in the actual field. The middle initial and suffix are in a light font. If you add the suffix there, the name will appear in various places in the chart such as the MRE was John Smith, Jr. but the middle initial (I would contend more important than the suffix for ID) doesn't. Only the first and last name show up on the scripts.
Ironically, when I view certain documents pertaining to the patient in other areas, the middle initial shows up.
I have never run into that situation, because generally even if I have a patient with a very common first and last name and the most common birthday of September 9th, it is enough to identify them.