Back in the 1970s, when synthetic engine oils (those based not on petroleum but on chemical base stocks such as polyalphaolefins) first became popular, they didn't always play well with the seals and gaskets in the car's engine. They could cause the seals to shrink in ways that petroleum-based oils did not, resulting in those messy oil leaks that would mysteriously appear in your car's parking space. Some people still fear that synthetic oil will cause leaks and so they continue to use petroleum-based oils instead.
These fears are largely unfounded. Oil manufacturers long ago learned to reformulate synthetic oil so that seal shrinkage doesn't occur. Still, there's a way in which synthetic oil can cause a leak, at least when you use it in an older car that's been operating for years on a petroleum-based oil. The synthetic oil can clean oil sludge off the seals that may actually have been blocking off tiny cracks in the seals, revealing leaks that have been there all along. This probably won't be a problem on newer cars, but if you're still driving a car that's more than, say, 15 years old, you might not want to make a sudden decision to switch to a synthetic oil.