If I trademark a name/logo this month and begin business with that name this month, but someone else has been using the same for 3 years but never registered it as a trademark, can I trademark it? Can that other person sue me for the trademark since they used the name first?
If someone else has been using it in commerce before you then their rights will predate yours. In the USA it comes down to "first in use," and you've acknowledged that someone else has started ahead of you. Even if you move forward and attempt to formally trademark it ahead of them.
I'd consult with an attorney on this before moving forward. If you intend to build a business and protect your IP (which you should), you should make sure that you are doing it the right way and not risking any infringement or other issues surrounding your right to use the name/logo.
Legal disclaimer: I am licensed to practice law in the state of Washington and the answer provided above is for general information purposes only and should not be relied on as specific legal advice. This answer does not form an attorney-client relationship. You should consult with an attorney of your choice to fully advise you about your legal rights and obligations.
Family Law Attorney
I agree with Attorney Camm. Even if the first user hasn't trademarked his logo, his right to use it supersedes yours and you can be sued. Consult with an IP attorney on how to avoid a lawsuit.
Please note that I am answering this question as a service through Avvo but not as your attorney and no attorney-client relationship is established by this posting. An attorney-client relationship can only be established through signing a Fee Agreement and paying the necessary advanced fees.
As long as they have been using the mark before you, then they can sue you, regardless of whether their mark is registered or not.
I agree with the previous answers, however it should be noted that common law trademark rights are limited to the geographic area in which the mark is used. The prior user of this non-registered mark could only enforce his/her/its rights against you in the geographic region in which the mark has been used.