I think you should define exactly to what privileges you refer rather than present us with a one shot squib regarding quarters. Quarters are a matter of what are available on a given post or station and how that might relate to a given billet.
I can assure you that no colonel is going to get assigned to Quarters 1 at Fort Myer.
I've known an Admiral, who commanded a carrier division out of South Boston Annex, was assigned duplex quarters at, then, Chelsea Naval Hospital all the way across town, with a Commander & his family on the other side of the wall while the hospital commander, a Captain, lived in a turn of the century mansion down by the river with a Captain's Row for the hospital's senior officers of 4-5 single family houses from the 1920's stretched along the river bank. Duplex was nice though, probably built in the 1930's, four stories, plenty of room.
A Colonel, moving into the right job, might get a large set of quarters which would equally please a General, but that is rarely a guarantee. You might want to consider that, by far, most general and flag officers and, yes, colonels and navy captains, do not live on a post or station but are, as it is known, "out on the economy." The reality is that usually specific jobs entitle one to quarters, for example, a colonel serving as chief of staff to a lieutenant general and the general, who rates quarters, wants his CoS ready and available, if not already planned for, the colonel will probably get the next set of quarters available.
On older posts and stations, some of these quarters look really nice but are maintenance hogs. The quarters on "Admiral's Row" on the naval Base in Norfolk were all built for the 1907 Jamestown Exposition and the Navy works very hard on their upkeep. I can tell you, though, from experience, that it is not unusual for a Captain to be assigned one of these as quarters, certainly not the largest and most visually imposing, but certainly more than ample. My father, then a captain circa 1960, had quarters on said "Admiral's Row" as did at least 2 other captains I can remember. One was the base commander, another was medical corps type, and my father was NavAirLant assistant operations officer . . . all living in quarters alongside the big shots. Evidently someone somewhere decided it was important to have these folks on base and near their posts.
If you are just Joe Colonel, another staff weenie of some sort, non-command type, and you want on post quarters, you can ask nicely if any will be available, but don't hold your breath. And they haven't allowed ranking out since the early 1900's so a colonel can't bump some major out of his quarters nor can a general bump the colonel lucky enough to get quarters.
