Graduate students with teaching or research assistantships are employees of the State of Connecticut. Income tax must be paid on both the federal and state level.
The University files federal declaration of money you have paid in tuition and fees. Because tuition is waived for TAs and RAs, only the student fees are reported. If awarded funds are received as a lump sum, they are reported on the form as well. In the absence of explanatory documentation, the government will assume that awarded funds (under their header of Scholarships and Grants) are used to pay tuition, when in fact you are using them for research expenditures.
Part-Year Residents of Connecticut
If you just moved to (or away from) Connecticut, your state income tax will require the CT-1040NR/PY form to determine the tax owed.