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Mr. Thompson’s support of McCain-Feingold and his chairmanship of a Senate probe of the 1996 Clinton-Gore re-election campaign are vulnerabilities, Mr. Bopp said.

Mr. Thompson’s Governmental Affairs Committee hearings were convened in 1997 to investigate foreign contributions to the Democratic campaign that were clearly illegal under federal law. The hearings instead ended up treating legal Republican donations as part of a bipartisan scandal, a perception eventually used to justify the 2002 law best known by the name of its Senate sponsors, Sen. John McCain, Arizona Republican, and Sen. Russ Feingold, Wisconsin Democrat.

“He should have concentrated on the illegal activities of the Clinton administration and the Democratic National Committee in the 1996 election,” said elections-law lawyer Cleta Mitchell, who advises conservative clients. “Instead, Thompson let the Democrats on his committee run roughshod over him, and the only thing that came out of the entire effort was the record the Supreme Court used to validate McCain-Feingold.”