Not exactly.
The Republican Contract With America was in 1994.
1998 saw House Republicans file impeachment charges against Bill Clinton.
The weakness didn't begin until after Democrats radicalized and criminalized politics after losing Bush v Gore in 2000.
Trent Lott was the Majority Leader in the Senate during Clinton's impeachment. After that, he folded during the power sharing deal with Democrats when Jim Jeffords switched sides. Prior to this, he was weak when dealing with Democrats during the prior lame-duck session when they briefly retook control.
After Lott resigned in 2002, Bill Frist became the Majority Leader, with Mitch McConnell as his deputy. This pairing of Frist and McConnell is generally when Republican weakness became metastasized. It has reached terminal levels this year.
-PJ
I trace it back to the first government shutdown, which I think was 95. Republicans turned all pussy after that.