The call came in at approximately 7:30 a.m.
The fire started on the fourth floor and there were flames and smoke showing when fire crews arrived.
Crews managed to knock the fire down quickly and all residents of the six family home were able to exit safely.
New Bedford Fire Chief Robert Frates believes it was an accidental mattress fire and does not believe there is anything suspicious.
As many as 25 firefighters were on the scene and the Red Cross was notified to help displaced families.