Apple has launched a replacement program for USB-C charging cables included with MacBooks sold through June of 2015, as well as those sold separately during the same time period. While Apple doesn't mention what portion of the cable is at fault, it says that affected cables may not charge correctly when connected to a MacBook:

A limited number of Apple USB-C charge cables that were included with MacBook computers through June 2015 may fail due to a design issue. As a result, your MacBook may not charge or only charge intermittently when it's connected to a power adapter with an affected cable.

Apple will provide a new, redesigned USB-C charge cable, free of charge, to all eligible customers. This program also covers Apple USB-C charge cables that were sold as a standalone accessory.

According to Apple's post, affected cables do not have a serial number listed next to the "Designed by Apple in California. Assembled in China" label on the cable. The redesigned cables do have a serial number next to the label.

Redesigned USB-C cable diagram

Apple says that it will proactively ship out new cables by the end of February to those who have registered valid addresses during product registration or through an online purchase through the Apple Store. All other MacBook owners that have affected cables can contact Apple Support via telephone, or visit an Apple retail store or Authorized Service Provider to get a replacement.

Source: Apple