I used to build winches for the theatre industry and I know that a properly sized cable should be a tight fit. Having said that, let me ask the obvious. Is there corrosion in the hole making it smaller? Was the cable cut with a high quality cable cutter leaving the cable unfrayed and the same diameter at the end as the balance of the cable?
If you do need to ream the hole larger it will likely be fine as there ought to be a couple of "safety wraps" that are the source of friction holding the cable which is always loaded (unless you run aground board down).
Hope that helps a bit?