One example being MIFARE used in the London underground, where they found a vulnerability in the encryption algorithm being used on the card (the cards are 'intelligent' in that they have a processor on board to perform the cryptography etc.). Once you can break the encryption, you can then set about making a clone.
With contactless bank cards, (which in the UK you can use to pay for the M6 toll), you could imagine people electronically picking your pocket, by relaying the signal from a transceiver near a cash point over RF to another transceiver which communicates with the card in your pocket. (hopefully they use distance bounding protocols to prevent this!)
Karsten Nohl (http://www.cs.virginia.edu/~kn5f/) has some fascinating papers on how they broke Crypto1 (the algorithm used by MIFARE). They took photos of the chip in the card using an optical microscope, after dissolving the card in acetone (They had to sand off each layer of the chip to get to the next)
They then applied image recognition to identify logic gates from their constituent transistors. And then looked for XOR gates, as they mention they're rarely used for anything other than crypto on this type of chip. Once they had found the area of the chip involved in cryptography, they converted the logic gates into the actual algorithm used, and then picked apart weaknesses with it.