I worked for and had a non-compete contract with a corporation with a name starting with World. If I created a organization that starts with same word but for completely different purposes, would that be infringement on my part?
Look at trademark type law first: Probably not an issue. If the word in question is truly as ubiquitous as "World", well, this is a common word, right? Even if Company has a federally registered mark in, for example, "World Wide Sports" (selling sports equipment), it should not be able to assert legal rights against someone else using "World Wide Pants" (selling television entertainment and "Top 10" lists). The idea here is that the "World Wide Sports" company, just because they have a registration in the full name "World Wide Sports", does not get to forbid the rest of us from using the common words "World", "World Wide", for completely different purposes.
Now, a caution here is words that are not common and are being used as marks. Take "Xerox" as an example. This was a completely made-up word at the time they began using it for a trademark. But now it is arguably a "famous" mark that immediately calls to the public's mind one thing only - the Xerox company's products. So someone trying to start a company called "Xerox Fishing Worms" could still be in trouble under a couple of legal theories that relate to "famous" trademarks.
Second question is your non-compete agreement. Note these are contracts governed by state law. And different states are literally all over the map (sorry for the bad pun) when it comes to what are, and what are not, permissible restrictions under non-compete agreements. Go back and read it very carefully. From your post, it sounds like you are not going to actually "compete" with them, since you say it is a completely different purpose. But you still may have promised to abide by other provisions (for example, a very strictly stated "non-use of names" clause) that could (on their face
) be legally binding upon you.
Then as your next step, if you find that you have agreed to any of these types of strict promises, the question for you to figure out under the laws of the relevant state is whether or not these promises are really
legally binding according to state law. It is not at all unusual that a non-compete agreement can be so out of whack with state law that a court will refuse to enforce parts or all of such an agreement.