The package with Biden's name on it was found at the facility in New Castle, Delaware on Thursday morning.
Late Tuesday night, the Secret Service intercepted a package addressed to Hillary Clinton before it reached her home in Westchester County, New York.
A picture of what is believed to be a pipe bomb sent to CNN headquarters on Wednesday. There's also been no immediate claim of responsibility.
"The #FBI has confirmed three additional packages, similar in appearance to the others - one in NY addressed to Robert DeNiro, and two in DE addressed to former Vice President Joseph Biden", the Bureau tweeted without further comment.
That person, who has a background in law enforcement, recalled seeing a similar package at the film center mailroom earlier and reported it to police.
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Leading Democrats called the bombs a risky outgrowth of coarsening political rhetoric promoted by Mr Trump, who also condemned the acts but blamed the media, his frequent foil, for much of the angry tone.
CNN Worldwide President Jeff Zucker condemned Trump for the rhetoric that seemingly contributes to continuing hostility toward the press. The package, now considered suspicious because of its similarities to others, was misaddressed and returned to the sender. "It has gotten so bad and hateful that is beyond description". It's unclear whether the return address was Wasserman Schultz's.
Investigators are now working in NY and other cities, in the postal system, and at the federal level, to determine who sent the packages - and why. Trump has repeatedly criticised CNN as "fake news" and disparaged the mainstream news media as an "enemy of the people". This time, the package, which appears to contain a pipe bomb, was found at the TriBeCa Grill, which is owned by Robert DeNiro, a vocal opponent of Donald Trump.
In addition to its appearance, the package also had the same return address as the others: that of Democratic Rep.
The bomb squad responded and removed the device, which was placed in an NYPD bomb vehicle to be taken to a safe area.
The US president said that "acts or threats of political violence have no place in the United States", adding that a "major federal investigation" is under way.
Senate Democratic Leader Chuck Schumer and House of Representatives top Democrat Nancy Pelosi said Trump's call for unity rang "hollow" because of his past statements condoning acts of violence.