Yes, it's possible you would get a federal refund, yet owe taxes to the state.
The federal tax rules are very different from state rules. Most states start with your federal return, then apply their own rules.
Some states have a fixed tax rate rather than the federal graduated rate. States have different standard deductions and personal exemptions and allow different credits from the federal government. For instance, as a student, you may have gotten a nice education credit from the IRS that wasn't allowed by your state.
The difference in result in your federal and state returns doesn't mean that they were in error. You should have copies of both documents. Read them carefully, and you may figure out where the difference came from. (And if you didn't get copies, ask for them!!)