In Connecticut the statute of limitations (deadline to file suit) is based upon the claims being made, not the court the case is being heard in. Determining what deadline exists should only be done after consulting with an attorney because specific case facts can alter what claims may or not apply, and the time to bring such claims.
In general, an action on any contract in writing “shall be brought but within six years after the right of action accrues” pursuant to Conn. Gen Stat Sec. 52-576. As the question says the contract involves home improvement the presumption is that it was in writing.
This is important because (in most circumstances) the contract must strictly comply with the Connecticut Home Improvement Act, Conn. Gen. St. § 20-418, et. seq., that regulates home improvement businesses and persons who offer, undertake, or perform any home improvement. A good overview (from a consumer perspective) of their rights under the Act can be found in my Article (Staff Pick) at http://www.avvo.com/legal-guides/ugc/protecting-connecticut-homeowners-from-home-improvement-pitfalls-1
In cases where the agreement does not comply with the Home Improvement Act the contract can be void, thereby negating the contractor’s right to compensation, and can provide the homeowner numerous defenses and independent causes of action against the contractor. Therefore, it’s advisable to consult a business law attorney prior to starting a lawsuit that has the potential to turn on the contractor if its agreement is not compliant.
For more information on the Small Claims process visit the Court website http://www.jud.ct.gov/Publications/CV045.pdf and their guidebook (http://www.jud.ct.gov/Publications/CV045.pdf)
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Disclaimer: The foregoing answer does not constitute legal advice, is provided for informational and educational purposes only for persons interested in the subject matter. Each situation is fact specific and may be subject to state specific laws. Without a comprehensive consultation and review of all the facts and documents at issue it is impossible to evaluate a legal problem fully. This answer does not create an attorney-client relationship.