Perhaps you found the seed potatoes you originally planted. They are always mushy when they come up.

Here is a link that might be useful: My potatoes and garden

Hopefully you did not plant whole potatoes, so I expect that you are experiencing bacterial soft rot (Erwinia carotovora subsp. carotovora). It is pretty widespread, but really manifests itself under certain growing conditions.

Here is a link that might be useful: Soft rot