Not necessarily. The word may be a common word (possibly in another language), the current owner may own the rights in another country, the current owner may have a business with the name and is holding the domain name for future use, etc. You should start by contacting the current owner, and ask whether they’d be willing to give you the domain, or sell it at reasonable price. If they decline without explanation or asks a high price, contact a lawyer.
Simply put, no it doesn’t give you a right to the name… unless someone deliberately buys the name with the intention of misleading potential visitors into thinking they are you, you have no claim over the combination of letters and numbers as a brand. This is why when I built my brand I bought up all the TLD’s that applied to me (COM / NET / ORG / BIZ / INFO / CO.UK / ORG.UK / EU), it gives me greater control scope over my brand. Trademark law only protects your brand in respect of what you offer as an entity (and it only protects it within the nations the trademark is applied within), it doesn’t give you the right to all instances of a words usage. If that were true then Apple would be able to claim ownership over all the fruit that bare it’s name
There is also something called common law trademark which means that if a name or word has been in use before the trademark was registered then they have common law rights to use that name.
I do agree, from first hand knowledge, that what Alex said was true that unless someone buys a domain with the intention of misleading visitors or reselling the domain to you at a profit, then they have the rights to use it.