Having a potential function and being conservative are equivalent (under some mild assumptions).
Specifically, if a (continuous) vector field is conservative on an open connected region then it has a potential function.
And "Yes" if a vector field fails to be conservative, it cannot have a potential function.
Here are some notes I posted for one of my classes a few years ago...
A few notes:
1) I didn't list all assumptions everywhere (for example, I wasn't careful to say that I'm assuming things are continuous where needed).
2) In the notes a vector field which possesses a potential function is called a "gradient" vector field.
3) The relevant theorem is on page 5.