Which makes it all the more problematic when O'Brien was referred to several times as a Lieutenant during his tenure on board the Enterprise. Memory-Alpha has the episodes and references.
It's possible that serving on a Starship allots you a rank and a commissioned benefits. I believe in DS9 Emissary there was a line about people who serve on Starbases are the people who weren't good enough to be assigned to a Starship. DS9 was full of NCO's. If O'Brien was really that good given his service on board the Phoenix and the Enterprise, a field commission promotion could be feasible