You can. To file a trademark you must either provide an example of an actual use in commerce, or an affidavit setting forth your intent-to-use the mark. Given that this mark has not been used for 30 years, you would need to file an intent-to-use affidavit, and follow that up with your line of t-shirts, etc. You will want to speak with an attorney to provide a clearance search , and to conduct the filing and prosecution.
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I agree you can protect the name and use it on shirts and the like, but just a word of caution: you might run into issues if you use images of the mascot and it resembles other mascots. Colleges have been known to sue high schools over mascots, such as Penn State suing Buna High School in texas over the lion/cougar. Tuskeegee Univeristy also uses Golden Tigers, so just be cautious if your tiger looks like someone else's.
You can obtain trademark protection for this name, but be careful. Before doing so, you need to retain counsel to conduct a "trademark search" to assure that you are free to use the trademark without violating rights of others. For example, you may not be able to use a Tiger in connection with this trademark if such use might violate trademark rights of other colleges or professional sports teams. Further, filing for trademark protection is a complicated matter---you need to decide whether to file an "intent to use" or "actual use" application, you need to decide how to register (the name or phrase, a logo with the name or phrase, or perhaps both), and you need to decide what classes to register. Then you will face trademark prosecution in the trademark office which will probably involve filing legal arguments to a trademark examiner who raises questions about your application in a "rejection". Thus, if you want to pursue this, you should retain trademark counsel to handle this for you. Otherwise, the trademark you get may not be worth the paper it is printed on.
In addition to the other Answers, you should keep in mind that the US Patent & Trademark (USPTO) office may refuse subject matter such as a words, slogans, designs, or logos as it may be deemed merely a decorative feature and does not identify and distinguish the applicant's goods, thus it fails to function as a trademark, according to the USPTO. This is known as the Refusal on Basis of Ornamentation and is fairly common in connection with clothing.
You may wish to speak with an attorney before or during the process of designing the goods in order to ensure the mark is used as a trademark.
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