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.
I think PayPal was first to be released before Bitcoin was released, though Im not quite sure about that. There are lots of people who trusts PayPal, so I dont see anyway for Bitcoin to really compete with it. I think we should just mind our own business here on bitcoin, while PayPal fans mind theirs.