Compass bearing accuracy:
1) Assumes you can walk or hike on a compass bearing accurate to a half or one degree. You need a proper sighting compass for this - and a mark on the ground some distance off to help keep you on track;
2) You have no metal objects about your person that will affect the compass;
3) There are no local magnetic anomalies.
If you are striving for this level of accuracy you also need accurate pace counting/distance measuring. Your pace length will vary according to terrain (up, down, steep, rocky, boggy, snow, chest deep bracken, wooded, tree roots, etc.), ambient light, your fitness and your tiredness, the load you're carrying, and who you're hiking with. Again if you need this level of accuracy you need to be aware of, or use, search patterns such as the expanding box search to find your target when you know or realise you've missed it.
To add a little less accuracy to the debate: I have sailed thousands of miles at sea in small boats. In a gentle seaway it is difficult to keep the boat pointed to an accuracy of even five degrees; in a large seaway ten to twenty degrees would not be unusual yet I still managed to make successful landfalls, including taking into account compass deviation (varies according to heading), leeway (approximated), tides and ocean currents (also approximated), and this before the advent of GPS. Even a position line from a sextant on a small boat can be several miles out.
Navigation is an art, not a science: the art of navigation is "Knowing you are not where you think you are". To this end you use all available tools: map/chart, compass, sextant/theodolite, GPS, and especially the Mark I Eyeball. Knowing what to expect (stream, ridge, fence) and when (five minutes, one hour) and ticking off the features as they appear is the way, as welshwhit and Tor helge suggest above. After all, you can see where you are. (I'm here!) A map shows much of what is around you - both visible and invisible, assuming you know where you are on the map.
It is also worth bearing in mind that a decent modern GPS, particularly with WAAS/EGNOS, will give you a position that is more accurate than than the map you are using, that is the terrain was surveyed at a lower level of accuracy some (many) years ago. OSGB produces some very reliable maps but even they 'fess up that "individual features on the map may only have been surveyed to a local accuracy of 7m (for 1:25000 scale maps)".
Consider also the scale of the map: 1mm on a 1:25k map is 25m; on 1:50k map it's 50m!
I've just returned from a trip which included hiking in temperate rainforest where the trail changed direction 10 times every 30 metres, and the trail would swing through up to 180 degrees as I traversed slopes via a switchback. It's not even worth trying to keep a DR plot going. I just kept note of time on the trail (well marked or blazed trails). Where there was doubt I left markers to indicate the route back if I needed to retrace my steps. I knew where I was - on the trail, about x hours in - and I had a vague idea where I was in relation to the surrounding topography. It was good enough.
It's important to be pragmatic with navigational accuracy.
IMHO.