Not being pushed to overtime at home by garbage like Washington. Find me any of the past ten National Champions tested to that extent by dreck like UW on their home field.
I won't limit it to home games, but here goes... National champions over the past 10 years with close games against teams without winning records.
The '04 USC Trojans
3 point win over Stanford (finished 4-7)
5 point win over UCLA (finished 6-6)
The '03 USC Trojans
None, but did lose to 8-6 California
The '03 LSU Tigers
None, but did lose to 8-5 Florida at home
The '02 Ohio State Buckeyes
4 point win over Cincinnati (finished 7-7)
11 point win over Northwestern (finished 3-9)
7 point win over Illinois (finished 5-7)
The '00 Oklahoma Sooners
5 point win over Oklahoma State (3-8)
The '99 Florida State Seminoles
3 point win over Clemson (6-6)
The '98 Tennessee Volunteers
8 point win over Auburn (3-8)
The '97 Michigan Wolverines
None, but close wins over mediocre (nearly .500) Notre Dame and Iowa teams at home
The '96 Florida Gators
7 point win over Vanderbilt (2-9)
Loss to Florida State
The '94 Nebraska Cornhuskers
10 point win over Oklahoma (6-6)