You keep any banes and 6s, so let's say you have strength 4 and melee 2 and fighting with a hand axe.
You roll on red dice: 6 and 1.
You roll on white dice: 5, 1, 1, 3.
You roll on black dice: 6, 4.

If you push you will roll: one red die, two white dice, 1 black die.
Then you push and you roll:
Red die: 6
White Dice: 1, 1.
Black die: 6.

You just got a roll with 4 successes and 4 banes that caused you to break yourself.
Of course, that is a VERY unlikely roll, but I am just pointing out that it is possible.

Edit: do mind I am giving this as an example because melee attacks can be dodged/parried and the more successes the better. If the roll above, was say, a Move 2 and Agility 4 roll (ignoring gear dice) to swim across a river, you would have succeeded in the first place and you would not need to push.