How can I test a three-way switch to determine if it's working?
1: You have a 3 way switch which has a traveler line. 
your odds of having a bad 3 way keeping the fixture from not working while possible would mean that both positions [Off and On] of the switch fail, or both switches have failed. Because when one switch is off the other line is working (either of the switches can turn the light on or off.)

2: Measure for voltage at the fixture wires. 


Turn one switch On and then Off while measuring do you have any voltage ?
Turn the other switch On and then Off while measuring do you have any voltage ?


I am going to post a wiring diagram of a 3 way switch lighting circuit so you can see how and where you need to place your leads when checking the switches. I would be measuring VOLTAGE.



There has been a comment on using the push terminals on the electrical switch or using the terminal screws - the push is faster and more likely why people use it and also if you have connected runs. 
Don't hook two wires on one Screw , use the push terminal to make that additional connection.

As for removing wires from the push terminals:
I use a 3.5mm Slotted Screw Driver mine is just like the one pictured below and I have no issues with it - of course I am experienced with push terminals and know that you must insert into the release mechanism and apply pressure on the release either by a slight twist of the screw driver (recommended) or using a prying motion to open the release.



You can obtain the tool from here :

https://www.crum.com/2307041/Product/3.5MM_RED_SLOTTED_SCREWDRIVER

I am not endorsing this vendor, I am posting their link because that is where I obtained the picture from. I have no affiliation with them and have never done business with them.