Some wells do dry up in drought although it's mostly dug wells or shallow wells that are less than 100 feet deep that are more prone to dry up during droughts.
If your well is drilled at least 100 feet or deeper and you have a good high water table then it's not very likely for the well to dry up even during drought.
Some wells have to be drilled 200 feet or more to reach water and some only need drilled 100 feet or less to reach water.
The shallower the well and the lower the water table the higher risk you have at your well running dry during a drought or from simply overusing the well.
My well is drilled 150 feet deep and my static water level is 10 feet and recovers really good.
Drilled wells are better than a dug well as drilled wells can reach deeper sources of groundwater which means that the water is less likely to be contaminated and it's less likely that the well would dry up in a drought.
Dug wells are okay but they are prone to drying up from drought if you don't get enough rain or precipitation and they are at higher risk of becoming contaminated.
When you drill a well you can drill to 100 feet or more and reach cleaner water and in most cases the deeper you drill the less chance you have of it drying out as there is usually more water the deeper you go.
My well is 150 feet deep and we get a lot of rain here and so it's very unlikely that the drilled well would go dry.
Even here I can dig down 2 feet and water fills the hole although it's dirty water.
So since our water table is at 2 feet down it means that my drilled well is unlikely to go dry anytime soon and we get plenty of rain.
I have had a dug well in other dryer locations and they went dry when he didn't get enough rain.
But my drilled well has never gone dry and unlikely would and I could probably get millions of gallons out of it a day if I had a pump that big that could draw that amount of water and it still would likely not go dry.
Wells that are for households and for drinking water should ideally be at least 100 feet or more and some locations you have to drill to 200 to 300 feet or even deeper to reach any water depending on how far down the water table is in your location.
Places that get a lot of rain and precipitation have a higher water table and you don't have to dig as deep but in dryer locations without enough rain or precipitation you usually have to drill down deeper to reach any water.