To illustrate the effects of precision on accuracy we will consider measurements made of a known true value and then imagine being able to adjust the width of the uncertainty interval while the mid point remains the same distance from the true value.
It is important to point out though that although increasing precision improves accuracy this improvement is not always of any practical benefit. This is explained in the following paragraphs.
However in practical situations the measuring instrument will never be perfectly calibrated and so the true value will be offset from the centre of the uncertainty interval.
Now if you reduce the uncertainty interval the limits will be pulled closer to the centre and as before the limit which is furthest from the true value will be pulled closer to it. So as before error is reduced and accuracy improved.
However from the diagram you can see that centre of the uncertainty interval will always be closer to the true value than the furthest limit is. This means that the error in the measurement will always be greater than the offset between the centre of the uncertainty interval and the true value. So if this offset itself would be considered too large an error then there is little point in improving the precision.
To illustrate this point consider the futility of taking extremely precise measurements from an instrument which has very large calibration errors!
To further illustrate the relevance of precision and accuracy the following diagrams show examples of two different measurements taken of the same true value.