Well using laser propulsion, we might be able to get a very lightweight probe up to 1/4 c using technology that isn't too far off.[0]
Coincidentally, the group working on this will be presenting some of their most recent work on this at 2:55 EST today. You can watch that live here[2].
The bigger deal is, How do you deal with the ablation problem at those speeds? Given the likely density of H in interstellar space, erosion is your primary problem.
Coincidentally, the group working on this will be presenting some of their most recent work on this at 2:55 EST today. You can watch that live here[2].
[0] http://www.deepspace.ucsb.edu/projects/directed-energy-inter... [1]https://www.nasa.gov/sites/default/files/atoms/files/2016_sy... [2]http://livestream.com/viewnow/NIAC2016