We present high-precision, model-independent, mass and radius measurements for 16 white dwarfs in detached eclipsing binaries and combine these with previously published data to test the theoretical white dwarf mass-radius relationship. We reach a mean precision of 2.4 per cent in mass and 2.7 per cent in radius, with our best measurements reaching a precision of 0.3 per cent in mass and 0.5 per cent in radius. We find excellent agreement between the measured and predicted radii across a wide range of masses and temperatures. We also find the radii of all white dwarfs with masses less than 0.48 M-circle dot to be fully consistent with helium core models, but they are on average 9 per cent larger than those of carbon-oxygen core models. In contrast, white dwarfs with masses larger than 0.52 M-circle dot all have radii consistent with carbon-oxygen core models. Moreover, we find that all but one of the white dwarfs in our sample have radii consistent with possessing thick surface hydrogen envelopes (10(-5) >= M-H/M-WD >= 10(-4)), implying that the surface hydrogen layers of these white dwarfs are not obviously affected by common envelope evolution.