Virtual scenario:
 
Average system. Mediocre PSU. Gamer. Decides to upgrade GC.  System doesn't take it anymore under GPU stress.
 
Differential diagnosis with limited equipment: PSU standalone test with PSU tester (eg fibrionic PSU tester) turns out OK. All voltages on all rails good. Confirmed by multimeter. DD: PSU overload when gaming (GPU stress too much for poor old PSU)
 
Constricted procedure: full system test. Measuring voltages on random unconnected molex rails. +5V and +12V. Observing voltages idle and stressed.
 
Question: Will the observed voltages drop along with the PSU not being able to deliver sufficient power via the ATX and AUX connectors?
 
Assumptions:
1: A modular high quality PSU would be able to deliver steady Voltages and powers on all but the failing/overloaded rail. Therefore one would not be able to observe voltage drops on other rails (eg molex cons).
2: The PSU is overloaded and  one would be able to observe voltage drops or spikes on other/all rails.  
 
Benefit: If one was able to observe before mentioned symptoms one could be able to restrict the failures to a weak or bad PSU. Therefore limiting other sources of failure. (memtests and burn ins not included)
 
Qualified input appreciated.
 
 

The different rails could show different voltages, depending on construction of the psu.  However, if one rail fails, it kicks the protection circuit and the psu shuts down.  This could be the condition you describe.
 
However, if your psu has active power factor correction (PFC), it may attempt to compensate for the voltage drop, and you could likely only see the psu fail without a voltage drop.
 
Other factors would be heat levels, age, and how hard you have pushed the psu before the upgrade.
 
As far as psu testers, go, you should understand that these are not LOAD testers.  Voltage and load tests are different.  You would need a LOAD tester to properly test the psu for the conditions you describe.
 
You would want to set up a scenario where the system fails, and check your voltages at the point of, or just before the failure.  You could test each rail of the psu in this fashion, to do a full analysis.