Can I repair a cut-through credit card?

I'm quite sure you will be able to recover "some data" from the magnet stripe - likely even all the data that's on there. However, you won't be able to withdraw any cash with a copy of that stripe because nowadays use the information on the chip, which is much safer as it is much harder - close to impossible, I would guess - to copy that. The only thing you could hope for is a store with an old terminal, but then you have to give your copy of the card to a person...

As for the chip, if you don't have the means of an intelligence agency or something like that, you won't be able to repair it. (The chip is not just a "data store" that is to be read but a device that can answer cryptographic challenges.) Same thing should apply for contactless payment.

Thanks for the comment: I'm referring to the situation in the European Union, namely EC-cards, which are most common over here.

For mag stripe, as answered by Thomas. But note that most countries implement EMV differently and may allow mag stripe purchase if the chip fails to read.

The actual chip area of an IC card is very small. (If you look carefully at a well-used card, you may be able to see the location of the actual chip.)

If the cut is not right through the chip, the chip can probably still be salvaged (e.g. by attaching wires to it).

The antenna for contactless payment will certainly be damanged by cutting, but it can normally be fixed if the chip is not damanged.

Obviously, the latter two require professional equipments and expertise, but criminals increasingly invest in these areas unfortunately.