Addai had failed the team physical, a league source told ESPNBoston.com Patriots contributor Field Yates.
The move comes as a mild surprise, with Addai signing a one-year contract in May that included a $75,000 signing bonus and $50,000 workout bonus. His veteran presence had been lauded by some of the team's younger backs in spring camps.
When asked about Addai's release on Thursday, coach Bill Belichick said simply, "It didn't work out."
In other moves, the team removed linebacker Brandon Spikes from the active/physically unable to perform list on Wednesday, a league source confirmed. Spikes had missed all offseason practices with an injury, but this likely signals he will be ready to practice on Thursday.
Reserve offensive lineman Nick McDonald was placed on the active/PUP list Wednesday. The reason for the move was not immediately known. McDonald had been a full participant in organized team activities and minicamp.
The 32-year old Shiancoe signed a one-year deal, according to a league source. It includes a $1.2 million base salary, $400,000 of which is guaranteed. Incentives can push the total value of the contract to $2 million.
Bulman is a native of Milton, Mass., and played at Boston College. The 29-year old had played for the Houston Texans since 2006.
ESPN Boston reporter Field Yates contributed to this report.