Usually when we think of cover-up tattoos we think of an old lover whose name was tattooed in a moment of madness on some part of the body (usually highly visible) only to be regretted after the relationship suddenly ended. This is somewhat of a cliche and most often not the case.
Generally a lot of people are presenting to tattoo studios for cover-up tattoos simply because they want to touch up an old faded tattoo or rework an old design that does not resonate with them any longer. It may be the original tattoo design was not the most professional job or is just starting to look a bit dated.
This can often happen when a small tattoo was done without considering the bigger picture. For me personally that meant getting a small kanji tattoo covered over with a bigger tattoo that meant more to me. There is nothing wrong with rethinking your ink and covering up with a bigger and better tattoo.
In the last few years there has been quite a bit of advancement with tattoo inks. This has led to a whole new variety of colors and inks that look brighter and better than the inks of old. In that sense many old tattoos can actually be made to look much better than what they did beforehand without taking away from the skill of the original artist of course 
Many artist also specialize in cover-up work so it is a good idea to check through their portfolios to see some of the cover-up tattoos they have done. Don’t settle for an artist who is content to cover up your old tattoo with heavy blackwork but is willing to use some creativity in integrating the old design with a newer better one.
Even if the original tattoo was black it is still possible to work the old design into a newer more colorful design with a little bit of imagination either by yourself or the artist who will probably have a better idea what will work and what will end up really dark and undefined. Remember it is your body so you decide what the finished outcome is.