I think it might be interesting to consolidate all of the examples we find of non-officer personnel within Starfleet in the 24th century in this thread.
TNG "Family"
Miles O'Brien, Chief Petty Officer
Worf's father was also a Chief Petty Officer
TNG "The Drumhead"
Simon Tarsis, Crewman 1st Class
Examples of non-officers in TNG+
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in TNG o'brien gets shown to be a lieutenant jg according to the pips on his collar, yet in DS9 he's referred to as chief petty officer or something similar. it seems like they never keep his rank consistent in the series, is there any explanation as to what the ranks are? 
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