I believe this is one of those things that benefits from scale. Theoretically, the larger you make the sail, the better the thrust to mass ratio you can achieve (even before calculating a better payload mass to sail mass ratio). With improved materials, we can make stronger and lighter sails and support structures, and this will in turn result in higher velocities by the time the vehicle has left the effective range of the sun. I think speeds truly approaching c are unlikely, but they can still achieve "really freaking fast".

But then new advanced materials could also change that, we're developing metamaterials with some fascinating properties, carbon nanotubes are just the tip of the iceberg. Who's to say that we couldn't some day achieve those speeds.