My CNC router at the program for mentally-handicapped people have steper motors similar to the one shown in the image. Can wires inside the motors go bad?
If too much current is run through them, yes.
Other possible problems include heating them above the curie temperature and permanently demagnetizing the not-so-permament-after-all magnets, and getting abrasive dust into them and damaging whatever moves.
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You can test the magnets by turning the axle of the motor. If the magnets are still okay, you will feel the typical stepping sensation.
When the magnets are heated, the motor will run smoothly. The Curie temperature of neodymium magnets is around 310-350°C, so the paint on the outside of the motor might also be scorched if overheated.
When your CNC table has two identical motors, you could try measuring the resistance of the windings of both motors. If all four windings have the same value, the problem could be in the driver modules.
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Oh… I thought magnet wire enamel would melt at a higher temperature than the curie temperature, but I was wrong — lots of it has a lower melting point than the magnet curie temperature. My memory for typical enamel melting point was of °F, not °C as I thought. Looks like the enamel would melt and the windings become shorts before the curie temperature was reached. On the other hand, the windings becoming shorts would not cool down the motor, so if the windings have shorted, good chance that the magnets have also lost their magnetism, and it fundamentally doesn’t matter because the motor is very dead either way. Also, at those temperatures, they might get into the range of melting the solder, too, so there could be some open rather than short circuits.
Smart diagnostic advice. 
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My experience with long used stepper motors is that the bearings wear which causes friction to increase(because of the side-loading)and the motors start making noise and skipping steps.
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What about wires coming out of motors?
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Those wires can break due to metal fatigue when they bend during operation
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I think that the problems are caused by the green terminal blocks that plug into the drivers. The wires can come off the terminals if they are not tightened properly or if the wires are not inserted properly into the terminals.
I used ferrules but I think the ones from Amazon are junk.
Or the terminal blocks. I had a problem like that with cheap chinese terminal blocks.
You can try to sqeeze the ferrules a bit more in the middle to make the tip a bit thicker. After that they might come loose but they don’t get out of the terminal.
Or fasten the terminals with a really good fitting screw driver and hold the terminal block between pliers.
While the motor windings can be burned or coils broken, these motors are pretty robust and take a good amount of abuse. Electronics and motor wiring harnesses are most likely to fail. The failure of the bearings can occur if shaft rotation by hand shows irregular behavior or seizes.
Having said that, if you explain the CNC malfunction you’re experiencing, we might offer some help.
Check that there is no black (burned plastic) on the green terminals.
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It might just be the part where the motor connects to the screw thingy that needs tightening. If the motor turns but the axis doesn’t move its probably those connections. Ive had this happen a couple times and then checked every single one to be sure they are tight.