Nope, I bought some "winery overrun" corks from MoreWine. They're 1 3/4 inch long.
It's like the part of the corker that fits over the top of the bottle is too short to fit both the cork and the bottle in there.
Am I supposed to push the cork halfway into the bottle by hand or something? That doesn't seem possible.
Here's a better picture:
EDIT: I had some old corks I've taken out of store-bought wine bottles, and I smashed them into my bottles by hand. They're only halfway in, but it'll have to do for now. I must be the stupidest person alive. I seriously can't figure out this corker :'(
The corks I bought are larger in diameter than the ones that came out of the store-bought wine. MoreWine didn't have any size differences in their descriptions, but they did say which corks NOT to use with a hand corker. These corks weren't listed.