Used to be no problem if you were just a local covers band if you happened to share a name with a band ten states away. Unfortunately, the internet has solved that for us forever, and two bands who twenty years ago might never have heard of each other can now have an ugly slanging match over who's the "Original" Saturday Morning.
I just went through this myself researching a band name for my new covers unit. In my old day job I used to do a lot of internet research, so here's some tips to check if your band name is already taken:
1) Google is your friend. Obvious, right? But a lot of people don't know how to properly use search engines. Just typing in "Saturday Morning" (for example) will get you a lot of junk links up top and not clue you in that there's a band out there with that name. Try typing in "(band name) +band +music". The plus sign makes emphasis of the word directly following - in other words, it shuffles pages with those words in it to the top of the search.
2) Use Facebook, MySpace, PureVolume, and ReverbNation as search engines. Any band worth its salt has a page on at least one of these networks. You may find a band in very close proximity to you using the name, which would be an even greater problem than one in Helsinki or Ouagadougou.
3) Use checkdomain.com to check to see if someone has reserved domains related to the band name. Two reasons for this: a) you might find a band that for whatever reason did not crop up in steps 1 and 2; b) it'll come in handy come website registration time.
4) Use one of those goofy band name generators to make up a name so silly that it's unlikely that anyone ever used it. (Only halfway kidding.)
If you do find out that you have a band name conflict, in this day and age of global village citizenship, the right thing to do is change yours, in my opinion. Hopefully you didn't already nail down any web addresses or permanent MySpace names. But it probably happens more often now than ever, if only because of the sheer amount of acts out there, and the internet making bands more visible than ever.
Good luck, good hunting.