There are a few good examples, depending on how one defines bipartisanship, but the best recent example came in 2006 - presumably what McConnell is referring to. In the last thirty years, there have been four debt limit increases which passed at least one of the House or the Senate with support from only one party.
Most recently, in February 2014, S.540 - Temporary Debt Limit Extension Act passed the House supported by 28 Republicans, but passed the Senate 55-43 without any votes from Republican Senators.
Earlier, in December 2009, H.R.4314 - To permit continued financing of Government operations passed the House without any bipartisan support, but was supported in the Senate by one Republican, George Voinovich.
In March 2006, H.J.Res.47 - Increasing the statutory limit on the public debt passed the Senate supported by only Republican votes, and was only possible due to the passage of H.Con.Res.95 in 2005, which passed both the House and the Senate without any Democrat voting in favour.
Finally, in 2004, S.2986 - A bill to amend title 31 of the United States Code to increase the public debt limit passed in the House without Democratic support, but was supported in the Senate by two Democratic Senators, Zell Miller of Georgia and John Breaux of Louisiana.