Do airlines set cabin pressure to a higher altitude than what the aircraft is capable of maintaining?
Most modern airliners have an automatic pressurization system that maintains cabin pressure at about 6000′ to 8000’.
Older airliners required pilot input and management to maintain adjust and properly set the automatic pressurization system and maintain comfortable cabin pressures. If the system was not set properly the cabin can have a higher cabin pressure than what is typically desirable. Although rare, it is possible to set a higher cabin altitude.
Also, aircraft can have a leaking door seal to one or more passenger/cargo doors that could cause a large amount of leaking. If the leak is larger than what the air system can bleed into the aircraft, then the cabin altitude could be higher than what is normally desirable.
Here is an image of the Boeing 727 environmental panel.
As you can see in the bottom right area there is a setting for Cruise Altitude (FLT ALT), a setting for Landing Altitude (LAND ALT) and a setting for Cabin Altitude (CAB ALT). The Flight Engineer would set these per the performance charts and system limitations for a particular flight.
The more modern systems do not have altitude settings. They simply have an Auto or Manual mode. On this Boeing 777 environmental panel the AUTO mode follows a preprogrammed set of takeoff, climb, cruise, descent and landing modes to maximize comfort and ease of operation.
Manual (MAN) mode is very basic and not intended for normal operations. As you can see it only has a basic landing altitude setting so the aircraft does not land pressurized.
Maintaining a comfortable low cabin altitude is the most desirable goal of all airlines. There is no monetary savings or gain by adjusting cabin altitudes higher or lower.