Legally, band names are pretty much like... brand names (no pun intended).
First of all a technicality, names are not copyrighted, they are trademarked and can also be registered.
Legal details of course depend from country to country, but in general, like with any business, if you can show that you have been operating a business or selling a product using a certain name, in theory (i.e. without taking into account the difficulties of going to court against a big shark that's ripping of your name) you can ascertain in court your ownership of the trademark and the right to exclusively use that name (within the same or similar lines of business). Ways of showing that you have been using the name prior to any others, are, in the case of bands, for example contracts with promoters, advertisements and published work. Specially if you have a record that's recognized by a professional author rights association as a work published under an artist's name, that's a pretty good protection for that name.
The safest way to protect the name, however, is to make it a registered trademark. I think that registering a trademark at country level in Europe is not too expensive, probably not either in the US at state level. Registering a trademark at European or world level is very expensive and it's a recurring expense (you have to have attorneys constantly looking at possible violations).
A cheap and easy way to have some level of protection nowadays is to create accounts/profiles for your band in every social network and online music platform you get your hands on (facebook, twitter, reverb nation, soundcloud, myspace, etc.). Not only these can be additional evidence in court if ever it comes to that, as ripping of your name will be deterred by the fact that the name is already being used in the platforms.
BTW, there are cases of relatively known bands with the same name, but usually in different time periods and operating mostly in different countries (an example is that of the bands both called 'Home' in the UK and in the US).