Paypal is the direct competitor of Bitcoin so not a chance in hell. Bitcoin is designed to be peer-to-peer effectively making PayPal services absolutely redundant if it became mainstream.
In the global marketplace, rivalries can be friendly for a day, there are many examples. Apple and Samsung look like competitors, they can act jointly. But I think Paypal is a finished company.
In the near future all the big companies will start accepting bitcoin as their payments. It is good that PayPal has started accepting bitcoin as their payments and they are trading in bitcoin because they know the importance of bitcoin and they have realized that bitcoin is the future currency and all the people in the world will use bitcoin as their legal and regular currency for their day to day needs.