* Distinctive: It must be different enough from other marks in the same field to avoid consumer confusion. Generic terms or descriptive terms are generally not considered distinctive enough. "Virginia" is a geographical term, not a distinctive brand.
* Not merely descriptive: A name like "Virginia Tech" is partly descriptive (it's in Virginia and has a technological focus) and therefore wouldn't be easily trademarked unless it acquired secondary meaning.
* Not generic: The name of a state, city, or common descriptive term is considered generic and cannot be trademarked.
However, there's a crucial exception: A name that has acquired secondary meaning *could* be trademarked. This means the name, even if initially descriptive or generic, has become so strongly associated with a particular source (a specific school, for example) that consumers automatically associate it with that source. This would require years of use and significant marketing effort to demonstrate the secondary meaning. Even then, the trademark would likely be limited to specific goods or services related to that school and its activities (e.g., merchandise with the school's logo, but not necessarily general educational services).
In short: While "Virginia" itself is not trademarkable, a specific logo incorporating the word "Virginia" or a unique and non-descriptive school name *might* be trademarkable after demonstrating secondary meaning, but this is a difficult and costly process. Most schools will rely on copyright protection for their logos and specific creative materials.