Z-axis does not work

Thanks for your fast response!
They are plugged separately, like this:

Ok, thanks! I'll try it.

Very strange...and you readjusted the vref potentiometer after splicing?

By any chance have you unplugged / plugged the stepper motors from the board with the power on?  This is a bad practice as it can blow the stepper chip...

I see the vibration...but why are ALL your motors cycling back and forth at the same time? 

Can you move the X and Y separately with no problem?

To rule out the driver chip, power down and try switching the Z motor plug for X or Y and try driving them that way.

Can you post a very clear straight-on picture of your RAMPS board showing all the connections?

OK...everything looks proper (btw, does the printer have a servo of some sort?)...

So, when you switched the plugs, the Z motors behaved the same way when driven by another driver?...then did the Z driver chip then drive the X (or Y) motor you switched with smoothly?

If so, put everything back and then separate the Z motors and try driving them one at a time...you may have a bad connection at the plug...it is unlikely to have two bad motors, especially since they are behaving the same.

One more thing to check would be to verify the motors coil pairing...take the plug off the board and jumper the green and black contacts together in the plug with a paper clip.  Then try to manually turn both motors.  They should both be very difficult to turn.  Then do the same thing with the red and blue wire contacts.

After that, aside from the vref adjustment, I'm afraid I am at a loss...

Actually, one more minor thing to check...you said you re-flashed your firmware...if you are using Marlin, there is a line in Configuration_adv.h that should be commented out:

Make sure that line has the backslashes in it...

If you don't mind splicing wires, just put both motors in parallel by splicing the same color wires together and then connect them to just one Z axis output on the board. Another poster had the same issue and this was his solution that worked. In a nut shell you combine both motors on one output. This is actually a common configuration.

Just make sure the color sequence at each motor is the same as well.

Well, I got the axis to work, thanks to you, IronMan!! There was a problem in the RAMPs also, but it just worked when I tried to use two separate drivers. I didn't even know that was possible, so thanks for that post. Well, all axis are working now, but I still have some problems with the extruding... But, this would need to be in another topic. Thank you one more time!