I had a secondary air monitor that would not reset with a BMW M54 engine. I tested every component multiple times, and replaced all hoses. This engine uses a solenoid vacuum valve on the intake manifold, and the vacuum operates a valve on the exhaust manifold.
The easiest way to test the valve system is with a MityVac and a 12V source. The vacuum gauge on the MityVac lets you monitor the operation in place, and the pump lets you test them individually for valve operation and leaks. Applying 12V to the solenoid valve should open it to the intake vacuum, with no power it should vent to the atmosphere. These rarely fail on their own, but may crack. The exhaust valve shouldn't leak when closed, and the diaphragm should hold vacuum when operated. These usually fail by not closing the whole way, allowing exhaust moisture into the main air pump which rusts to failure.
In my case the whole system was working correctly, but the oxygen sensors were slow. Not slow enough to raise an error during regular engine operation, but too slow during warm-up for the ECU to be certain that the secondary air system was pumping air. Replacing both upstream and downstream oxygen sensors cleared the monitor. (The cheap oxygen sensors caused their own errors a few months later, but at least they served to diagnose this tough problem.)