Ian
This is a complicated subject with several answers depending on circumstances. In practice, I suppose that a really expert navigator can achieve an accuracy of 2° when walking on a bearing but the occasions in the UK when that degree of accuracy is required normally only apply when navigating in restricted visibility, and especially when walking alone.
Factors that make it difficult to walk accurately on a bearing are wind, slopes, and broken terrain which tend to make the walker drift off-course. If you navigate to an accuracy of 4° (which in my opinion is more realistic) you will be off-course by 70 metres after 1000 metres, and if 6° you will be 105 metres off-course.
Rather than walking a couple of kilometres relying solely on a compass bearing, it is often better to to navigate to a series of readily identifiable features which you can tick off on your map. It may take longer and involve taking several bearings but is more reliable. Accurate walking on a bearing may be supplemented by taking backbearings and techniques such as pacing, timing and expanding square searches..
If in pathless terrain I was walking to a bridge over a watercourse, I would probably aim off by as much as 10° to ensure that I knew in which direction to turn when I reached the watercourse.
Hugh
I grow old...I grow old, I shall wear the bottom of my trousers rolled. T.S.Eliot