I regularly treat people who want cover-ups but their artist sends them to me first to remove the top layers of ink. The way this technique works is by removing a tattoo layer by layer. Obviously, if the top layers are free of ink then a cover-up will be much easier and more successful.
I had a patient come to me recently who had had a cover-up which had not worked at all. The name she was trying to hide was very clearly visible through the surrounding ink. I am now working on just the name to pare it back.