Sen. Dianne FeinsteinDianne FeinsteinSenate sets date for hearings on Sessions's attorney general nomination Senators move to protect 'Dreamers' Mika Brzezinski: Clinton camp wanted me off the air MORE (D-Calif.) on Tuesday became the fifth Senate Democrat to formally call for a rewrite of ObamaCare so people who like their health insurance can keep it.

Feinstein announced that she would join Sen. Mary LandrieuMary LandrieuFive unanswered questions after Trump's upset victory Pavlich: O’Keefe a true journalist Trump’s implosion could cost GOP in Louisiana Senate race MORE's bill, the Keeping the Affordable Care Act Promise Act.

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"The Affordable Care Act is a good law, but it is not perfect," she said. "I believe the Landrieu bill is a commonsense fix that will protect individuals in the private insurance market from being forced to change their insurance plan. I hope Congress moves quickly to enact it."

Late last week, Sens. Kay HaganKay HaganGOP senator floats retiring over gridlock 2016’s battle for the Senate: A shifting map The untold stories of the 2016 battle for the Senate MORE (D-N.C.) and Mark PryorMark PryorCotton pitches anti-Democrat message to SC delegation Ex-Sen. Kay Hagan joins lobby firm Top Democrats are no advocates for DC statehood MORE (D-Ark.) joined the bill. Those two and Landrieu are in tight re-election races next year, but Feinstein and the fifth cosponsor, Sen. Joe ManchinJoe ManchinSenate passes stopgap funding bill, averting shutdown Senate advances funding measure, avoiding shutdown Senate sets date for hearings on Sessions's attorney general nomination MORE (D-W.Va.), are not up for re-election until 2018.

Feinstein said she is joining the bill after receiving thousands of calls and letters from angry constituents who may lose their health coverage.

"Since the beginning of September, I have received 30,842 calls, emails and letters from Californians, many of whom are very distressed by cancellations of their insurance policies and who are facing increased out-of-pocket costs," she said.

Feinstein's support is just the latest blow to President Obama, who last week was forced to acknowledge that millions of people are being told their insurance plans would be canceled because they don't meet ObamaCare's new standards.

Those cancelation notices go against Obama's explicit and repeated promise that if people like their insurance plan, they can keep it under the law. Feinstein said Congress must ensure that this promise is kept.

"Too many Americans are struggling to make ends meet," she said. "We must ensure that in our effort to reform the health care system, we do not allow unintended consequences to go unaddressed."

Her support is a sign that support for a legislative fix is growing even among Democrats. That factor could end up forcing Obama to accept some changes to the law, although White House Spokesman Jay Carney indicated today that Obama is leaning more toward finding ways to make it more affordable for people to use new plans that meet ObamaCare's standards.

At the same time, however, the House is expected to vote Friday on GOP legislation that is similar to Landrieu's, and some Democrats are likely to support that bill.