“I’m not happy with the Supreme Court,” President Donald J. Trump said on Jan. 6, 2021. “They love to rule against me.”

His assessment of the court, in a speech delivered outside the White House urging his supporters to march on the Capitol, had a substantial element of truth in it.

Other parts of the speech were laced with fury and lies, and the Colorado Supreme Court cited some of those passages on Tuesday as evidence that Mr. Trump has engaged in insurrection and was ineligible to hold office again.

But Mr. Trump’s reflections on the U.S. Supreme Court in the speech, freighted with grievance and accusations of disloyalty, captured not only his perspective but also an inescapable reality. A fundamentally conservative court, with a six-justice majority of Republican appointees that includes three named by Mr. Trump himself, has not been particularly receptive to his arguments.

Indeed, the Trump administration had the worst Supreme Court record of any since at least the Roosevelt administration, according to data developed by Lee Epstein and Rebecca L. Brown, law professors at the University of Southern California, for an article in Presidential Studies Quarterly.

“Whether Trump’s poor performance speaks to the court’s view of him and his administration or to the justices’ increasing willingness to check executive authority, we can’t say,” the two professors wrote in an email. “Either way, though, the data suggest a bumpy road for Trump in cases implicating presidential power.”

Now another series of Trump cases are at the court or on its threshold: one on whether he enjoys absolute immunity from prosecution, another on the viability of a central charge in the federal election-interference case and the third, from Colorado, on whether he was barred from another term under the 14th Amendment.

The cases pose distinct legal questions, but earlier decisions suggest they could divide the court’s conservative wing along a surprising fault line: Mr. Trump’s appointees have been less likely to vote for him in some politically charged cases than Justice Clarence Thomas, who was appointed by the first President Bush, and Justice Samuel A. Alito Jr., who was appointed by the second one.

In his speech at the Ellipse on Jan. 6, Mr. Trump spoke ruefully about his three appointees: Justices Neil M. Gorsuch, Brett M. Kavanaugh and Amy Coney Barrett, suggesting that they had betrayed him to establish their independence.

“I picked three people,” he said. “I fought like hell for them.”

Mr. Trump said his nominees had abandoned him, blaming his losses on the justices’ eagerness to participate in Washington social life and to assert their independence from the charge that “they’re my puppets.”

He added: “And now the only way they can get out of that because they hate that it’s not good in the social circuit. And the only way they get out is to rule against Trump. So let’s rule against Trump. And they do that.”

Mr. Trump has criticized Chief Justice John G. Roberts Jr. on similar grounds. When the chief justice cast the decisive vote to save the Affordable Care Act in 2012, Mr. Trump wrote on Twitter that “I guess @JusticeRoberts wanted to be a part of Georgetown society more than anyone knew,” citing a fake handle. During his presidential campaign, Mr. Trump called the chief justice “an absolute disaster.”

When he spoke on Jan. 6, Mr. Trump was probably thinking of the stinging loss the Supreme Court had just handed him weeks before, rejecting a lawsuit by Texas that had asked the court to throw out the election results in four battleground states.

Before the ruling, Mr. Trump said he expected to prevail in the Supreme Court, after rushing Justice Barrett onto the court in October 2020 in part in the hope that she would vote in Mr. Trump’s favor in election disputes.

“I think this will end up in the Supreme Court,” Mr. Trump said of the election a few days after Justice Ruth Bader Ginsburg’s death that September. “And I think it’s very important that we have nine justices.”

After the ruling, Mr. Trump weighed in on Twitter. “The Supreme Court really let us down,” he said. “No Wisdom, No Courage!”

The ruling in the Texas case was not quite unanimous. Justice Alito, joined by Justice Thomas, issued a brief statement on a technical point.

Those same two justices were the only dissenters in a pair of cases in 2020 on access to Mr. Trump’s tax and business records, which had been sought by a New York prosecutor and a House committee.

The general trend continued after Mr. Trump left office. In 2022, the court refused to block the release of White House records concerning the Jan. 6 attack on the Capitol, effectively rejecting Mr. Trump’s claim of executive privilege. The court’s order let stand an appeals court ruling that Mr. Trump’s desire to maintain the confidentiality of internal White House communications was outweighed by the need for a full accounting of the attack and the disruption of the certification of the 2020 electoral count.

Only Justice Thomas noted a dissent. His participation in the case, despite his wife Virginia Thomas’s own efforts to overturn the election, drew harsh criticism.

Mr. Trump’s rocky record at the court offers only hints about how the justices will approach the cases already before them and on the horizon. His claim of absolute immunity appears vulnerable, based on other decisions from the court on the scope of presidential power.

The case examining whether one of the federal statutes relied on by the special counsel in the federal election-interference case, which makes it a crime to corruptly obstruct an official proceeding, does not directly involve Mr. Trump, though the court’s ruling could undermine two of the charges against him.

The justices have been skeptical of broad interpretations of federal criminal laws, and the arguments in the case will doubtless involve close parsing of the statute’s text.

The case that is hardest to assess is the one from Colorado, involving as it does a host of novel questions about the meaning of an almost entirely untested clause of the 14th Amendment, one that could bar Mr. Trump from the presidency. The case is not yet at the Supreme Court, but it is almost certain to arrive in the coming days.

Guy-Uriel E. Charles, a law professor at Harvard, said the justices would have to act.

“The Supreme Court is a contested entity, but it is the only institution that can weigh in and try to address this problem, which needs a national resolution,” he said. “There has been some loss of faith in the court, but even people who are deeply antagonistic to it believe it needs to step in.”