Why does insufficient data lead to random errors?
More datapoints (if everything is done correctly) lead to more trustworthy results. With less datapoints, the possible (statistical) error grows, until at one point, when you only have as many datapoints as you have parameters, it looks like you have no error at all. (That is of course stupid, you should always have derived the error range of the individual data points.)

Now what is "insufficient"? Is that when the statistical error grows too large for your feeling? Or when you only have two points to draw a straight line through? Or when you only have two points to draw a parabola? That's your prerogative.

(note that I have also above used the word "error" in the same sloppy way your books seem to do. The "errors" are in your data, and they have a systematic component (hopefully small) and a distribution. The derived values (e.g. the slope of a linear regression) have a confidence interval.)