Hillary Clinton’s email team stiff-armed Congress on Tuesday when her chief tech staffer defied a subpoena and the two contractors she paid to wipe her server clean refused to testify, citing their right to remain silent.

Despite Mrs. Clinton’s urging last year that he give full testimony, Bryan Pagliano, the campaign operative she hired at the State Department and then had run her server, refused to attend the hearing of the House Oversight and Government Reform Committee.

Two other contractors for Platte River Networks, the Colorado-based company that used BleachBit to delete all of Mrs. Clinton’s remaining emails in March 2015, just weeks after Congress issued a subpoena for them, did appear but refused all questions, invoking their Fifth Amendment right against self-incrimination.

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Only one witness, former Clinton Foundation and Clinton family employee Justin Cooper, did testify.

Mr. Cooper acknowledged that the server he helped Mr. Pagliano run faced brute-force hack attempts with “some frequency.” Mr. Cooper also said he had access to the server, even without security clearances that would qualify him to handle the kind of top-secret information Mrs. Clinton was exchanging through the server.

Still, his revelations were overshadowed by the other three employees’ refusal to testify — adding heft to the growing questions surrounding Mrs. Clinton’s secrecy and her presidential campaign.

“Anyone watching today’s hearing in Congress knows that Hillary Clinton’s actions are far more corrupt than we even imagined,” Donald Trump, Mrs. Clinton’s Republican presidential opponent, said at a rally in Iowa. “This is a far bigger scandal than Watergate ever was.”

Committee Chairman Jason Chaffetz, Utah Republican, said there will be consequences for Mr. Pagliano, but he declined to say what punishment he is considering.

“When you are served a subpoena from the United States Congress, that is not optional,” Mr. Chaffetz said.

Mr. Pagliano’s refusal to appear was the most dramatic show of defiance Tuesday, but it wasn’t the only one. Paul Combetta and Bill Thornton, employed by Platte River, appeared but refused to answer questions, citing their Fifth Amendment right.

In one bizarre exchange, Rep. Elijah E. Cummings of Maryland, the ranking Democrat on the committee, asked Mr. Combetta and Mr. Thornton whether they intended to assert their right to refuse to answer each question. Both refused to answer even that question, insisting on their right to remain silent.

Platte River employees erased all of Mrs. Clinton’s emails in late March 2015, just weeks after The New York Times publicly revealed the existence of her secret account and after the congressional Benghazi probe issued an order demanding that all of her emails be preserved. Platte River used a tool that made the messages unrecoverable.

The FBI cleared Mrs. Clinton of legal jeopardy over the mishandling of classified information. The agency said Mrs. Clinton did risk national security but wasn’t sophisticated enough to understand the markings on the documents nor the technological bungling in which she was engaged.

But Mr. Chaffetz and other members of Congress have asked for follow-up investigations into whether Mrs. Clinton lied to Congress about her emails and whether she and her aides were guilty of obstruction by having her server wiped clean in defiance of the preservation order.

Mr. Cooper testified that his involvement with the server had ended by that time. He said that as an employee of the Clinton family, he helped buy and set up the first version of the email server and set up the clintonemail.com domain, though he said it was chiefly run by Mr. Pagliano.

He said only Mrs. Clinton, her daughter, Chelsea, and her top personal aide, Huma Abedin, had accounts at that domain.

He confirmed that hackers regularly tried to break into the system and that the attempts picked up particularly in the last 2½ years that Mrs. Clinton was heading the State Department.

Brute-force attacks are attempts to gain access to an account or server by repeated quick attempts to produce valid login credentials.

“We had developed systems to tamp these down,” Mr. Cooper said.

The FBI said attempts were made to access Mrs. Clinton’s account, but the agency could not determine whether the account had been compromised because of the sophistication of enemy hackers and the low-level technology of the Clinton server.

Mr. Cooper said he could not recall what level of encryption or security was attached to the server but did not believe it had two-factor authentication.

He said he lacked security clearance at the time he was helping maintain Mrs. Clinton’s server. He said he didn’t regularly see messages from Mrs. Clinton unless one was forwarded to him to print.

Mr. Cooper said he didn’t believe Mrs. Clinton set up the server to try to avoid open records laws.

Congressional Democrats blasted the hearing as a political charade designed to hurt Mrs. Clinton’s presidential campaign.

Some countered with their own accusations against Mr. Trump, asking the Justice Department to investigate whether he attempted to bribe Florida Attorney General Pam Bondi, who was reportedly considering joining a lawsuit against Trump University at the time the Trump foundation gave her campaign a donation.

“After receiving these funds, Mrs. Bondi declined to further investigate Mr. Trump’s business interests,” Rep. John Conyers Jr. and fellow Democrats on the House Judiciary Committee said in their letter to Attorney General Loretta E. Lynch.

“This fact pattern indicates that these payments may have influenced Mrs. Bondi’s official decision not to participate in litigation against Mr. Trump. A number of criminal statutes would appear to be implicated by this course of conduct,” the lawmakers said.