Full disclosure: I transcribe the lottery winning numbers from the Idaho State Lottery website into my newspaper, so I read the lottery results every day I work.
The really big lotteries like Powerball and Mega Millions have either one of two machines. In a one-machine game, there are balls from 1 to...oh, 69 (the current number of balls in the Powerball white ball machine) and all the balls come out of it. In a two-ball machine, the second machine has, say, 26 different-colored balls (once again, the current configuration of the Powerball red ball, or Powerball, machine) and one ball comes out of it. The Powerball can match one of the five white balls, but all five white balls will be unique. Idaho runs several medium-size games that also work like this.
Small lotteries like Idaho's Pick 3, Washington's Daily Game or North Carolina's Pick 4 have one machine for each ball drawn, and each machine has ten balls in it - zero to nine. Since there are as many copies of each figure as there are numbers in the game, it's completely possible (and really somewhat common) that all the balls could have the same figure on them. I checked the Idaho Pick 3 numbers; in the last 20 draws there were seven with a repeated number. It's unusual, but not unheard of, for all the balls to have the same number on them.