In most cases, an air compressor is designed to start with atmospheric pressure and then through stages increase the pressure to some known output pressure.
The design takes into account the amount of cooling between stages and the hp requirements to provide a balanced 1st ,2nd, 3rd. and 4th stage so the pressure on the crank shaft is evenly distributed. IF you consider a car engine with 3 each 3 inch diameter pistons moving 3 inch stroke, and then make the 4th piston 4 inch in diameter with a 3 inch stroke you are going to have something come apart rather quickly.
In a 4 stage compressor, each piston gets successive smaller with the same stroke length to keep the pressure on the crankshaft exactly equal and the unit in "balance". Also the cooling fins are designed to bleed off the excess temperature between cylinders.
IF you increase the intake pressure from 1 atmosphere to 2 atmospheres you have doubled the volume of the first stage and now the HP to pump this increased volume must also be increased considerably. With the increased volume also comes much greater heat of compression and the cooling surface of the inter-coolers is now "not sufficient" and the compressor overheats.
In the Walter Kidde compressors manufactured for the military - some had pressurized inlets to compensate for lower atmospheric pressure at high altitudes. They also had longer cooling coils to reduce the excess heat between stages.
Many years ago I attempted to increase the compressor output by pressurizing the inlet and ONLY succeeded in burning up motors and compressors. The most I could get away with - was adding 2-3 PSI to the inlet cover and the results were not worth the minimum benefit.
No big bloody calculations needed to determine what you might get away with.
Jim Shelden