The tattoo issue is one of the most often discussed topics on AN. Tattoos are becoming much more common, and a lot of people, including myself, do not care if their nurse, doctor etc. have visible tattoos, even if they are covered in tattoos. There are many people, however, who do not want a caregiver with tattoos, and I know managers and supervisors who will not hire people with tattoos, particularly for higher lever and management level jobs, even if there is not hospital policy barring tattoos. Is that fair? Not really, but that is the way it goes.
I tend to agree with what the PP said--perhaps it was your manager giving you a hit that if you want to advance, tattoos may hold you back. Again, personally, I don't think they should, but they may.