It is difficult to say there is a theoretical limit. The amount of hydrogen is directly related to the current you can flow through your membrane. The more current you flow through the membrane, the greater the amount of heat you will need to reject. (This is simple Joule heating due to electrical resistance.) Most PEM might tolerate up to say 120C before they start breaking down.

So, in theory, you could run at very high currents for short periods of time but that would limit the usefulness of the electrolyzer. Ideally, an electrolyzer runs continuously at outputs where you do not risk damage to the membrane.

The advantage to PEM electrolyzers is that you can run with pure dionized water on as electrolyte on the cathode and simply flow excess water as a coolant heat transfer medium.

Having said all that, for a solid state PEM, 1A/cm^2 is easily in reach for continuous operation. While it will depend on physical factors (heat transfer) from the cell, you may be able to obtain perhaps up to 5A/cm^2 for brief periods. The more amps the more volts required to push the electrons. Regardless, you will always have losses.

At the end of the day, expect efficiencies in the high 70s, perhaps at lower currents 80%. Most of my electrolyzer experience has been with alkaline electrolysis where low 70%s was the norm.

Bottom line: Keep your membrane cool!