They try to keep the effective altitude around 5-6K feet, as you found, for general comfort. I recall seeing someplace that FAA regs allow up to 8K feet, which might occur if a plane is operating near its maximum altitude (around 40K feet for modern jetliners), and some people can develop altitude sickness at 7-8K altitude so running at the limit is frowned upon. Normal cruising altitude is more like 33-36K so the differential with pressurization results in 5-6K.

That doesn't stop the air from being pretty stale. Depends on what kind of plane. 787 is specifically designed to both pressurize better, so it can maintain 5-6K cabin altitude at 40K and pull in more fresh air than many older planes. 757s are about the worst - something like 1/2 or more of the air is recirculated, supposedly for fuel economy reasons (??). For modern planes, the service ceiling is usually the point where cabin altitude reaches the legal limit.

Running near the ceiling is also a problem for performance. Great for fuel economy and running over (some of) the weather, but the window between stall and overspeed is very small which limits the ability to respond to things. I was on a flight back from Chicago one night where I was dozing with the ATC channel on (the other choices weren't attractive, especially the movie), and at one point heard ATC ask our flight to speed up a bit; the pilot responded that he couldn't - was already at VMax - and needed to get lower than the 40K we were cruising at to comply. Eventually we were moved down to 36K where in order to go faster.

(Minor aside into trains: the Chinese train to Tibet supplies oxygen for people who can't handle the near-15K ft altitude reached along the line.)