There are varying quality of both set screws and allen wrenches. The lower quality ones are made of softer metal and are more prone to stripping. It has been my experience that set screws are usually hard material whilst allen wrenches may be a softer metal, especially cheap ones that are shipped with the likes of faucet kits.
You may want to inspect your allen wrench. If the corners of the hex on the end have been rounded off this may be what is preventing you from removing the set screws. This would particularly apply if you have been using the same tool repeatedly over years. Try using a new allen wrench that is made of hard heat treated metal.
Try to inspect the set screw if possible. If the hex slot is so stripped that a decent quality allen wrench cannot get a bite in the slot then this is an indicator that the screw is probably a soft metal one.
Here are some possible alternatives you could try to remove the stripped out set screw. I've used these techniques various times over the years.
Get an allen wrench that is one size up from the nominal size of the slot in the set screw. Grind off the flats on the end of the wrench at a slight angle to taper the end so the very end approaches the slot size in the screw. Then force the wrench into the screw and hold it in tight while trying to turn it. The new bite may be all it takes to remove the set screw.
Use a small flat blade screw driver that has a hardened tip. Grind or file off the edges of the blade to a taper that would just fit into the slot of the set screw. It may also be necessary to thin the blade some depending on the screw driver you started with. Then jam the tip into the set screw slot and turn to try to loosen the screw.
Note with both of these techniques it is necessary that the corners of the modified tool are sharp so they can bite into the set screw slot. It will be more effective on a softer metal set screw. If you do manage to get the set screws out then make sure to plan replacing them with new ones.