First, don't ask police what the law is. You will get conflicting answers if you ask two or more officers. Their knowledge of gun laws varies, especially when it involves gray areas and minutia. When it involves guns their knee jerk reaction is to say "no, not legal" even when it is legal to do so.
As already pointed out, it is 100% legal, provided you can legally possess the firearm in MA and your final destination, to fly out of Bradly Airport with firearms following the airline's specific rules. CT has several carveouts for doing so which have been previously mentioned. In addition to what was already indicate for CT laws and FOPA, there was a U.S. Department of Justice memo publicly issued years ago (mainly due to NY and NJ's continual arrest of people flying through their airports with firearms) indicating so long as one follows FOPA they can transport firearms through airports in their checked luggage following airport/airline rules.
If someone has a link to that memo please post it, not finding it anymore in a quick search. Edit: Here is a link to that 2005 DoJ Letter:
https://www.handgunlaw.us/documents/doj_doc_nyc_air.pdf
PS: If your plane diverts to, or connects through, a ban state like NY or NJ before continuing on to your destination,
DO NOT pickup the luggage containing the firearms. Let the airlines retain possession of the luggage so they can transport it to the destination airport. Where people typically get arrested in NY and NJ is picking up their luggage containing the firearm, leaving the airport
and remaining in NY or NJ, then returning to the airport for their connecting flight and getting arrested upon checking with their firearms (usually a handgun).