Bennett said in a phone interview that Lugar's defeat should not be portrayed solely as "Tea Party vs. Lugar and the Tea Party won."
"It would be giving them more credit than they deserve," Bennett said.
Specter believes that Hatch, elected in 1976 with Lugar, should be on notice. Hatch was forced into a June primary by former state senator Dan Liljenquist, who is backed by Tea Party supporters such as the group FreedomWorks.
Hatch has "moved deftly to the right and has made a very persuasive argument on seniority" in the Senate, Specter said. "He may escape but it's taken a heavy toll on his voting record."
"If they think this means, 'OK, let's double down on efforts to attack Sen. Hatch,' all that will do is drive up his favorables and make him even stronger in the primary," Bennett said. "Utahns are angry at outside groups trying to dictate how they should vote."
(See also related CNN and NationalJournal stories, Reihan Salam blog post, and Philip Klein op-ed.)