First, we are in the realm of trademark law here. The fact that the two businesses are in different classes of goods/services makes a big difference. Think Delta Airlines vs Delta faucets.
That said, some companies may not see it that way of course. I have a client currently being opposed (USPTO) by RIM because they use the word "berry" in their name even though they are not doing anything related to that industry. And RIM , despite that fact that we disagree, has a right to oppose if they wish. So you alwyays need to consider that.
Here. it does not seem likely that you will face a problem because one must be using a trademark in order to enforce it and they are out of business (does not mean that they are not still using it in some way nor that they have no claim here, just that it is more likely).
You should make sure you conduct the proper trademark due diligence because it is often the mark holders you never see comong that create the biggest issue and in the US trademark holders have rights under what we call the "common law" as well.
I will link you to some helpful info below and welcome you to contact me if you want to discuss further.
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I searched the trademark name "Thirsty Turtle" on the Secretary of State Direct Access Web Site and the query returned "There are no records which match your query." The State Direct Access Web Site reflects that the Articles of Incorporation of THE THIRSTY TURTLE, INC. were filed on Feb 5, 1990 and the corporation forfeited its existence on Feb 18, 1998 for failure to pay franchise taxes. You should check the federal trademark registration to see if it is maintained by another entity.
Legal disclaimer: John Bonica is licensed to practice law only in Texas. His response is not legal advice and does not create an attorney/client relationship. The response is only intended to provide general information. The question may not include significant and important facts that would change the response. You should confer with a local attorney for competent legal advice.
There is a "Thirsty Turtle Beach Club" not too far up the road from my house in Illinois. I doubt they would cause you any trouble, but I suspect there are probably other "Thirsty Turtles" out there that might. The point is, whether or not your now closed local Thirsty Turtle still has any trademark rights in the name, that isn't the begining and end of it. You need to do a thorough clearance search to rule out any other companies using the same, or similar name, which could pretty much be any company with "thirsty" or "turtle" in their name. You really need to do a thorough trademark clearance search and have the results reviewed by a trademark attorney before you commit to any name, whether "Thirsty Turtle" or something else.