If the other name is registered to the company you have no chance. By being registered it should be displaying the little ® or ™ next to it.

The company I work for had been trading under a 3 letter name for over 20 years until someone in America actually registered it. We had to cease using that name or face action.

By registered I mean the name not the actual company. If its not registered specifically to them you can both use it. But in that case if you register it as a trademark first then they will have to stop using it.