One problem related to engine belts and pulleys has been reported for the 2013 BMW 528. The most recently reported issues are listed below.
The car had 73,000 miles on it. I was driving home from work in stop and go city traffic when suddenly, the engine went into limp mode (as it is designed to do after it detects engine problems) and indicators lit up saying that I needed to pull over as the car was inoperable. I was in the left lane about to merge onto a toll road, but now found myself trying to get the car safety over three lanes of rush hour traffic to a gas station. This I managed to do while the car was making load noises and not providing adequate acceleration. After having it towed to a BMW specialist I was told the engine's timing system was off. I was also told the timing belt fails often with this particular BMW engine. The common cause seems to be a (unknown at least to me) plastic part within that warps from stress and then causes said part to break apart within the engine. I was told after the engine was disassembled that plastic parts were found inside the engine. After replacement of that belt, I'm now told that the engine still has major problems. It now might require a new engine altogether. If after 73,000 miles a BMW needs a new engine, it suggests to me that NHTSA should do further research into this major engine problem and issue a recall to make these cars safe.