Its been very cold here the last little while very unusual for this area. The other night our water stopped working, We checked the pump and the pipes that are down in the basement and we know that they are not frozen, we also have heat tape on the pipes. We called a plumber who came out and confirmed that none of the pipes in our basement were frozen. Our neighbour who we share a well with also has no water my question is, is it possible for a well to freeze?

The Pipes in the well (near the surface and twoards the houses) will freeze LONG before the actual well would freeze.

What kind of well, dug well or drilled well? How deep?

More than likely the buried lines leading to the well.