There are several failure modes for logic gates and without further context for your test, it is hard to provide a more detailed procedure.
That being said, if you have access to a logic analyzer, you can monitor all the inputs and outputs and see in real time the operation.
Common failure modes are:
power supply dips that cause the IC to see a logic level change on the input without there actually being on and changing the output.
Drawing too much current from the output and causing the output to seem to work part of the time.
Input setup and hold times not being met or held correctly for the output to stabilize or hold the output stable.
Static damage that can cause one gate to change randomly or in connection to one of the other gates.
Output overloading on the entire IC. - Some IC's can only tolerate a total output of the chip, so at one point the output (loading; driving other inputs) from all the gates in total from the chip can be limited.
Speed, trying to change inputs too quickly for the output to reflect (propagation time)
The most common problem I have encountered in breadboard setups is power supply connections to the chips if they draw lots of power (the old 74xx stuff, then 74LSxx type stuff) some had significant power requirements and the 5 volts would show rail supply of 3v at the IC during fast switching.
I hope you get the idea that you can probe the input and outputs at static conditions but the device can appear to fail in real (fast) operation. It is not so easy without the proper tools to catch faults. Even older Logic analyzers (my personal HP is 15+ years old) have capture modes that catch glitches quicker than 500 pico seconds. So you really need either a digital scope or logic analyzer to catch some logic faults.