From an ESPN article on wacky moments in the draft.
In the first round of the 2002 draft, the Cowboys were attempting to trade down to select Oklahoma safety Roy Williams. Cowboys owner-general manager Jerry Jones made the trade with the Chiefs, and Dallas was poised to select Williams with the eighth overall pick, but Jones lost track of time.
"Jerry went to sleep at the wheel," former Cowboys scouting director Larry Lacewell said. "And all the sudden someone shouts that we only have 10 seconds left."
According to Lacewell, Jones' son, Stephen, dived across the table to call in the Williams pick and the Cowboys ended up with about a second to spare.
"That's as close as I ever saw us come to losing one," Lacewell said.
Nice. Get Jerry some Red Bulls before the draft.
Here's another one.
The Cowboys drafted Mississippi Valley State wide receiver David McDaniels in the second round of the 1968 draft based on his blazing speed. The only problem was that he couldn't run away from anyone during his first training camp.
Sensing something was wrong, the Cowboys asked him to run another 40-yard dash and he came in at 4.73 seconds, a bad time for a wide receiver. A quick investigation revealed that he had inadvertently been measured in the 38-yard dash at his college track. The Cowboys quickly traded McDaniels to the Bears, a team that also had an inaccurate time for the player, for Mike Ditka.
I think that trade worked out pretty well for the Cowboys.