Let's say you're opening a bottle of wine with a corkscrew and the cork breaks before its out. It's possible to get a broken cork out of a wine bottle. Even if you can't quite get the cork out, you can still get the wine out.
Unscrew the corkscrew from the cork.
Try screwing the corkscrew into the cork again. Screw it in about a half-dozen times until it feels like it has a good grip.
Pull the cork out gently. If you're lucky it should come out like a whole cork. If it doesn't work after a couple tries, or it breaks up, go on to the next step.
Push the broken cork into the bottle. Use the handle of a wood spoon or screwdriver or something like that. Do it gently so the wine doesn't splash.
Place a filter like a cheesecloth or a coffee filter over a decanter or pitcher or another container that can hold the wine.
Fasten the filter with string or a rubber band.
Pour the wine through the filter slowly. The filter will catch any pieces of cork that are floating in the wine.
Drink up. You'll want to finish it because the cork is still stuck in the bottle.