Even the best system is likely to have some potential errors. Examples are:
1. The most obvious are errors in making and recording the observations, which near the solstice must be made to an accuracy related to a very small fraction of the sun's visible diameter.
2. The instant of the solstice may occur just before mid-night. Since the sunrise on the next day is closer to the solstice instant than the sunrise on the true day of the solstice, an error in one day will occur if the solstice is based on the sunrise.
3. Ignoring the fact that the sun does not rise, nor set, in a 90 degree vertical direction, except at the equator. At all other latitudes it rises (sets) at an angle from the vertical which is dependent upon your latitude. This requires that you observe the solstice at the near instant when it's diameter is on the horizon - unless you have a shadowing surface (rock, mountain, etc.) that has the angle necessary for you latitude. Click on the menu item for Heel Stone Angle for a detailed discussion of this concept.
4. Lack of precision in placing the observers eye.
If you are observing the sun rise at Machu Pichu where the mountain that nearly shadows the sun may be a mile or more away it is not very critical if you move about a few feet while making the observation.
However, for the observers at Stonehenge observing the sun climbing up the angled "heel-stone" from a location between two guide stones located in the Stonehenge circle, a horizontal movement of only a fraction of an inch will cause an angular displacement enough to mis-calculate the solstice by a full day.
The calculations to support this conclusion are as follows:
The distance to the heel stone from the center of the Stonehenge stone ring is 80 yards (240 feet).
Sine 0.5 arc minutes = 0.000145 = horizontal movement in feet / 240 feet
Horizontal movement to cause an error of 1/2 arc minutes = 0.000145 x 240 x 12 inches/ft = 0.4 inches.
Since a crowd of individuals can all watch at once it is obvious that their eyes can be in different horizontal, as well as vertical, positions with respect to the heel-stone. Thus it should not be surprising if different observers come up with slightly different conclusions as to the exact day of the solstice.
By contrast, the tunnel at Newgrange uses a roof box to cast a shadow 19 meters away at the back of the tunnel. In this case the position of the observer's eye is unimportant, and the movement of the shadow for the 0.5 arc minutes is 0.000145 x 19 meters x 100 cm/meter = 0.28 cm or about 0.1 inch - which is perhaps a just visible movement over one day.
Of course, perhaps the ancient ones were not as concerned as we might be today and were happy to merely be able to observe a period of days in which the solstice occurred.