My roommates and I wanted to add another roommate to our lease. We now have three people in a three bedroom apartment. I called the landlord to inquiry if we could add a fourth roommate and was told that would not be possible. The landlord said there is a specific law in Massachusetts that states that more than three unrelated people cannot be living in the apartment because it is a three bedroom apartment. Does that sound right? I have never heard of such a law and would love to know if this is true.

There are local housing, zoning and building codes, state and Federal laws.
Contact your local buildling and zoning department, state housing office and HUD. Even if no such law exists, the landlord is not obligated to allow you to have another roommate added.

Ask the landlord to cite the law so you can verify it. But it is very possible such a law exists.