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A quick glance at any table of typical chemical shifts will reveal that the chemical shifts of protons $(\delta_\mathrm H)$ tend to correlate pretty well with the chemical shifts of the carbons to which they are attached $(\delta_\mathrm C)$. This correlation is frequently taught in introductory organic chemistry and the rule of thumb

$$\delta_\mathrm C \approx 20 \cdot \delta_\mathrm H,$$

although simplistic, actually holds up fairly well. To illustrate the point further, here is a diagram taken from the textbook by Silverstein et al. on organic spectroscopy:

Comparison of proton and carbon chemical shifts Image taken from: Silverstein, R. M.; Webster, F. X.; Kiemle, D. J. Spectrometric Identification of Organic Compounds, 7th ed.; Wiley: Hoboken, NJ, 2005, p 207.

To the beginner such a correlation might seem intuitive. However, delving deeper into NMR theory, one learns that the $\ce{^{13}C}$ chemical shifts are generally dictated by paramagnetic shielding, while $\ce{^{1}H}$ chemical shifts are generally dictated by diamagnetic shielding and effects due to neighbouring groups. See, for example, this question: NMR chemical shift range of different elements.

In light of this, how much should we make out of the general correlation seen? i.e., is it best understood to be a pure coincidence, or is there more to it than that?

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    $\begingroup$ I've never actually been taught that rule of thumb regarding dH shifts and dC shifts. Learnt something new today. THANKS! $\endgroup$ – Hazinga Aug 10 '17 at 20:36
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    $\begingroup$ I don't have the understanding to actually answer, but the conclusion has been drawn before and was definitely pointed out during my undergrad tutorials. Seems to be a coincidence that holds up okay for simple things. See J. Chem. Educ. 1991, 68, 284. An old article, with no explanation, but does offer an empirical conversion based on a training set of 53 (complex-ish) molecules. $\endgroup$ – NotEvans. Aug 19 '17 at 18:04
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Maybe it would help to comprehend the coincidental changes in 1H and 13C chemical shift by understanding the origin of chemical shift. In both cases, very crudely speaking, the influence of ring currents and electronegative elements/groups cause a downfield (away from zero) shift. The factor of 20 quoted is a function of the number of electrons (usually outer ones) involved in forming the bond. So the trend will also be true for other elements e.g. 15N where aromatic N compounds i.e. pyridine have a larger downfield shift.

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Apart from other specific effects, chemical shift is a function of electron density around the nucleus, so it is reasonable that proton shifts roughly correlate with those of the attached carbon.

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